Since C is Hausdorff and separable it follows that C has cardinality equal to 2 < sup > ℵ < sub > 0 </ sub ></ sup > — the same cardinality as the cardinality of the reals.

Since its models must have cardinality at least κ, the downward part of this theorem guarantees the existence of a model N which has cardinality exactly κ.

Since every set is a subset of this latter class, and every cardinality is the cardinality of a set ( by definition!

Since Af are linearly independent functions and the exponential function has no zeros, these R functions Af, form a basis for the space of solutions.

Since we are assuming circular motion, let r ( t ) = R · u < sub > r </ sub >, where R is a constant ( the radius of the circle ) and u < sub > r </ sub > is the unit vector pointing from the origin to the point mass.

Since 1974 with the publication of Elizabeth A. R. Brown's The Tyranny of a Construct, and Susan Reynolds ' Fiefs and Vassals ( 1994 ), there has been ongoing inconclusive discussion among medieval historians as to whether feudalism is a useful construct for understanding medieval society.

Since the 1970s, when Elizabeth A. R. Brown published The Tyranny of a Construct ( 1974 ), many have re-examined the evidence and concluded that feudalism is an unworkable term and should be removed entirely from scholarly and educational discussion, or at least used only with severe qualification and warning.

Since the core flux is proportional to the applied voltage, the iron loss can be represented by a resistance R < sub > C </ sub > in parallel with the ideal transformer.

Since the 1980s, prominent cigarette manufacturers such as Philip Morris and R. J. Reynolds developed fire-safe cigarettes but did not market them.

Since the 90s, Tangerine Dream have also recorded cover versions of Jimi Hendrix ' Purple Haze ( first on 220 Volt Live ) and The Beatles ' Eleanor Rigby, Back in the U. S. S. R., Tomorrow Never Knows, and " Norwegian Wood ".

Since the late 1960s other songwriter's award has been presented for genre-specific categories including Grammy Award for Best Country Song, Grammy Award for Best Rock Song, Grammy Award for Best R & B Song and most recently in Grammy Award for Best Rap Song and Grammy Award for Best Gospel Song.

Since the mid-1960s, Jones has sung nearly every form of popular music – pop, rock, R & B, show tunes, country, dance, soul and gospel – and sold over 100 million records.

Since a one sided maximal ideal A is not necessarily two-sided, the quotient R / A is not necessarily a ring, but it is a simple module over R. If R has a unique maximal right ideal, then R is known as a local ring, and the maximal right ideal is also the unique maximal left and unique maximal two-sided ideal of the ring, and is in fact the Jacobson radical J ( R ).

Since 1990, a fixed conventional value R < sub > K-90 </ sub > is used in resistance calibrations worldwide.

Since T is uncountable, this subset of R must be uncountable.

The transfer principle states that true first order statements about R are also valid in * R. For example, the commutative law of addition, x + y = y + x, holds for the hyperreals just as it does for the reals ; since R is a real closed field, so is * R. Since for all integers n, one also has for all hyperintegers H. The transfer principle for ultrapowers is a consequence of Łoś ' theorem of 1955.

Since 1990, a fixed conventional value R < sub > K-90 </ sub > is accepted.

Since < sup > </ sup > the bulk density of a solid chemical element is strongly related to its molar mass ( usually about 3 R per mole, as noted above ), there exists noticeable inverse correlation between a solid ’ s density and its specific heat capacity on a per-mass basis.

Since Led Zeppelin dissolved in 1980 with the death of Bonham, Jones has collaborated with a number of artists, including Diamanda Galás, R. E. M., Jars of Clay, Heart, Ben E. King, Peter Gabriel, Foo Fighters, Lenny Kravitz, Cinderella, The Mission, La Fura dels Baus, Brian Eno, the Butthole Surfers and Uncle Earl.

Since 2007, France Telecom R & D is also known as Orange Labs, a global network of R & D entities.

Since n grows faster than log < sub > 2 </ sub >( n ), there must exist a value n < sub > 0 </ sub > such that

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 >.

The proof of this fact relies on an algorithm which, given the first n digits of Ω, solves Turing's halting problem for programs of length up to n. Since the halting problem is undecidable, Ω can not be computed.

Since every equivalence relation over X corresponds to a partition of X, and vice versa, the number of possible equivalence relations on X equals the number of distinct partitions of X, which is the nth Bell number B < sub > n </ sub >:

Since the proving of the theorem, efficient algorithms have been found for 4-coloring maps requiring only O ( n < sup > 2 </ sup >) time, where n is the number of vertices.

Since it is a balanced tree, it guarantees insertion, search and delete to be O ( log n ) in time, where n is the total number of elements in the tree.

Since there are n!

Since the leftmost divisor bit zeroed every input bit it touched, when this process ends the only bits in the input row that can be nonzero are the n bits at the right-hand end of the row.

* Omission of the high-order bit of the divisor polynomial: Since the high-order bit is always 1, and since an n-bit CRC must be defined by an ( n + 1 )- bit divisor which overflows an n-bit register, some writers assume that it is unnecessary to mention the divisor's high-order bit.

Since 1953, " binomi < u > n </ u > al nomenclature " is the technically correct term in zoology.

Since this quantity goes to infinity as n increases, we expect that every large even integer has not just one representation as the sum of two primes, but in fact has very many such representations.

If n is not a prime power, then every Sylow subgroup is proper, and, by Sylow's Third Theorem, we know that the number of Sylow p-subgroups of a group of order n is equal to 1 modulo p and divides n. Since 1 is the only such number, the Sylow p-subgroup is unique, and therefore it is normal.

Since for each n this removes intervals adding up to at most 1 / 2 < sup > n + 1 </ sup >, the nowhere dense set remaining after all such intervals have been removed has measure of at least 1 / 2 ( in fact just over 0. 535 ... because of overlaps ) and so in a sense represents the majority of the ambient space.

Since every element of C < sub > n </ sub > generates a cyclic subgroup and the subgroups of C < sub > n </ sub > are of the form C < sub > d </ sub > where d | n, the formula follows.

Since the Supreme Court's decision of that year this is more doubtful ; ;

( Since the time-span of the nation-state coincides roughly with the separate existence of the United States as an independent entity, it is perhaps natural for Americans to think of the nation as representative of the highest form of order, something permanent and unchanging.

Since it is not far from Viareggio, he will visit Puccini's house, as he never fails to do, to pay his respects to the memory of the composer of La Boheme, which he considers one of Puccini's masterpieces.

Since attack serves to stimulate interest in broadcasts, I added to my opening statement a sentence in which I claimed that German youth seemed to lack the enthusiasm which is a necessary ingredient of anger, and might be classified as uninterested and bored rather than angry.

Since the air is a continuum, the network of communication remains intact regardless of the positions or motions of the points ( the people ) in the net.

Since the hazards of poor communication are so great, p can be justified as a habitable site only on the basis of unusual productivity such as is made available by a waterfall for milling purposes, a mine, or a sugar maple camp.

Since the difficulty of drawing the net is great, we will merely discuss it.

Since a civilizational crisis involves also a crisis in private interests and in the ruling class, reaction is normally found among those who feel themselves to be among the ruling class.

Since civilizational change is the most difficult to perceive and analyze, it seldom is given adequate attention.

Since the slogans have little application to reality and are sanctimonious to boot, the applause is faint even in areas of the world where we should expect to find the greatest affection for free government.

Since more is known about Quiney than about any other acquaintance of Shakespeare in Stratford, his career may be followed to its sudden end in 1602.

Since the great flood of these dystopias has appeared only in the last twelve years, it seems fairly reasonable to assume that the chief impetus was the 1949 publication of Nineteen Eighty-Four, an assumption which is supported by the frequent echoes of such details as Room 101, along with education by conditioning from Brave New World, a book to which science-fiction writers may well have returned with new interest after reading the more powerful Orwell dystopia.

Since Laos is of no more purely military value to Moscow itself than it is to Washington, this approach might be expected to head off Mr. Khrushchev for the moment.

Since little is known about autism, and almost nothing has been written for the layman, we'd like to share one experienced mother's comments.

Since the work is done by hand, the only limitation, it is said, `` is that of human conception ''.

Since the apparatus is new, it requires experimentation and changes in technique.

Since the obvious is not always true, the Republican National Committee wisely analyzed its defeat of last autumn and finds that it occurred, as suspected, in the larger cities.

Since he has just shown who is top dog, he may not be ready to receive this highest honor in the gift of the Soviet people.

Since appeals to morality, to humanity, and to sanity have had such small effect, perhaps our last recourse is the deterrent example.

Since the goal of our international planners is a World Government, this Atlantic Community would mark a giant step in that direction for, once American economic autonomy is absorbed, a larger grouping is a question of time.