for all smooth L < sup > 2 </ sup > functions f, g. Since smooth functions are dense in L < sup > 2 </ sup >, this defines the adjoint on a dense subset of L < sup > 2 </ sup >: P < sup >*</ sup > is a densely-defined operator.

Since functors are required to preserve composition, this says that the linear map corresponding to a sewn together morphism is just the composition of the linear map for each piece.

Since the homotopy category of CW-complexes is equivalent to the localization of the category of all topological spaces at the weak homotopy equivalences, the theorem can equivalently be stated for functors on a category defined in this way.

Since the circulating thyroid hormones are the amino acids thyroxine and tri-iodothyronine ( cf. Section C ), it is clear that some mechanism must exist in the thyroid gland for their release from proteins before secretion.

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

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 it is possible for such things not to exist, there must be some time at which these things did not in fact exist.

Since then Mao's peasant revolutionary vision and so-called " continued revolution under the dictatorship of the proletariat " stipulated that class enemies continued to exist even though the socialist revolution seemed to be complete, giving way to the Cultural Revolution.

Since Carmichael numbers exist, this primality test cannot be relied upon to prove the primality of a number, although it can still be used to prove a number is composite.

Since thousands of different types of wood products manufacturing plants exist around the globe, and may be integrated ( lumber, plywood, paper, etc.

Since then many clubs have admitted women, but some all-male clubs still exist.

Since organic compounds often exist as mixtures, a variety of techniques have also been developed to assess purity, especially important being chromatography techniques such as HPLC and gas chromatography.

Since evil has no positive reality of its own, it cannot be caused to exist, and so God cannot be held responsible for causing it to exist.

Since primitive worlds exist near technological worlds, primitive weapons are also typically included, such as swords, shields, pikes, bows, and so on.

Since most physicists would consider the statement of the underlying rules to suffice as the definition of a " theory of everything ", most physicists argue that Gödel's Theorem does not mean that a ToE cannot exist.

Since the late 1950s, aerospace engineers have used the term " unobtainium " when referring to unusual or costly materials, or when theoretically considering a material perfect for their needs in all respects, except that it does not exist.

Since none have been proven to exist already then the drive is impossible to construct, even if the metric is physically meaningful.

Since they exist, we cannot simply ignore them, for once we include the interaction between the electron and the electromagnetic field, any electron placed in a positive-energy eigenstate would decay into negative-energy eigenstates of successively lower energy by emitting excess energy in the form of photons.

Since a clear archetype for determining guilt does not exist, the criminal justice system must rely on rulings handed down by juries.

Since the supremum and infimum of an unbounded set of real numbers may not exist ( the reals are not a complete lattice ), it is convenient to consider sequences in the affinely extended real number system: we add the positive and negative infinities to the real line to give the complete totally ordered set, which is a complete lattice.

Since the shields are not perfect conductors, electric fields can exist inside the shield, thus allowing radiating electromagnetic fields to go through the shield.

Since physics needs to talk about numbers in offering any of its explanations, then numbers must exist.

Since the meaning of the existential quantifier in formal notation is usually explained in terms of its equivalence to English expressions such as " there is " and " there exist ", and since these English expressions are not reliably ontologically committing, it comes to seem that we cannot be sure of our theory's ontological commitments even after it has been regimented into a canonical formulation.

Since they are typically small, zooplankton can respond rapidly to increases in phytoplankton abundance, for instance, during the spring bloom.

Since Interstate 64 and the AA Highway both converge in Grayson, the northern part of town has an abundance of truck stops and fast-food establishments.

Since this dark green paint was in abundance on the railroad, it was the color used on the passenger cars.

Since then, Webster Groves ' tree-lined streets and abundance of single family homes have continued to attract people to the area as a " great place to live, work and play ," not solely for the wealthy commuter suburb that early developers envisioned but for families that cut across all socioeconomic lines.

Since the 2000 U. S. Census which allowed people to select more than one racial group, " Millennials " in abundance have asserted their right to have all their heritages respected, counted and acknowledged.

Since the sun has been fusing hydrogen into helium throughout its lifetime, the present day abundance of helium in the core can be used to infer the age of the sun, using numerical models of stellar evolution applied to the Sun ( Standard solar model ).

Since frugivores eat a lot of fruit they are highly dependent on the abundance and nutritional composition of fruits.

Since the tuckahoe was found in great abundance in this area, members of various New Jersey Lenape tribes would visit this location to collect them.

The abundance of iron, measured as equals, is only 3. 5 % of the abundance found at the surface of the Sun .< ref > Since 10 < sup >– 1. 45 </ sup > = 0. 035 .</ ref > The cluster shows evidence of being enriched by the elements generated through the s-process in massive stars and Type II supernovae.

Relative to the Sun, the abundance of iron in the cluster is given by = – 2. 29 dex, which equates to only 0. 5 % of the solar abundance .< ref > Since 10 < sup >– 2. 29 </ sup > = 0. 00513 .</ ref > This puts the estimated age range for the cluster at, or roughly the age of the Universe.

Since 4 % can be expressed as. 04, gives the amount of alcohol in terms of ml — which, when divided by, shows the number of units.

Since then it has been eroding, creating dramatic incised valleys, and providing the sediment that gives the Yellow River its yellow color and that causes the flooding of the river in the lower reaches ( hence the river's nickname ' China's sorrow ').

Since the tucked shape is the most compact, it gives the most control over rotational speed, and dives in this position are easier to perform.

Since the larger of the two numbers is reduced, repeating this process gives successively smaller numbers until one of them is zero.

Since the 2. 4 GHz band is heavily used to the point of being crowded, using the relatively unused 5 GHz band gives 802. 11a a significant advantage.

Since 2 < sup >( q − 1 )</ sup > ≡ 1 ( mod q ), raising both sides of the congruence to the power x gives 2 < sup >( q − 1 ) x </ sup > ≡ 1, and since 2 < sup > p </ sup > ≡ 1 ( mod q ), raising both sides of the congruence to the power k gives 2 < sup > kp </ sup > ≡ 1.

Since Eddie was headed deeper into addiction ( at the hands of his brother Henry ) or prison ( at the hands of the government ), or worse ( at the hands of his drug lord ), he decides to throw his lot in with Roland, although with deep misgivings that he occasionally gives vent to in the form of angry outbursts.

From the 1990s to the present, the majority of communications from “ one fixed location to another have moved by fiber .” Since these fiber communications are “ wired ,” U. S. law “ gives them greater protection .”

Since constitutional law cannot be altered by state legislatures, this direct democracy component gives the people an automatic superiority and sovereignty, over representative government ( Magelby, 1984 ).

Since then, the Mission has been repaired and restored, and has one of the best-preserved interiors ( which gives one of the best examples of old mission life ).

Since the pressure on top of the engine is ambient, this means that base gives no overall thrust ( but it also means that this part of the nozzle doesn't lose thrust by forming a partial vacuum, thus the base part of the nozzle can be ignored at low altitude ).

Since Ferdinando is still doing nothing, she spits in his face in front of the entire town, which gives him all that he needed: an open insult to the family's honor due to his wife's elopement.

Since 1822, rather than ordinary money, the Sovereign gives out Maundy coins, which are specially minted 1, 2, 3 and 4 penny pieces, and are legal tender.

Since the determinant of a unitary matrix is a complex number with norm 1, the determinant gives a group homomorphism

Since F and g are differentiable, the chain rule gives

Since 2002, the Sun has run an approximately annual contest called " Page 3 Idol " that gives amateur models an opportunity to win Page Three modelling contracts.

Since every formula in the antecedent ( the left side ) must be true to conclude the truth of at least one formula in the succedent ( the right side ), adding formulas to either side results in a weaker sequent, while removing them from either side gives a stronger one.

Since 4 and 6 are congruent modulo 2, a function defined on the integers modulo 2 must give the same output when the input is 6 that it gives when the input is 4.

Since Pappus gives somewhat full particulars of its propositions, this text has also seen efforts to restore it, not only by P. Fermat ( Oeuvres, i., 1891, pp. 3 – 51 ) and F. Schooten ( Leiden, 1656 ) but also, most successfully of all, by R. Simson ( Glasgow, 1749 ).

A simple version of this runs as follows: Since the brain in a vat gives and receives exactly the same impulses as it would if it were in a skull, and since these are its only way of interacting with its environment, then it is not possible to tell, from the perspective of that brain, whether it is in a skull or a vat.

Since her retirement in 1999, she has stayed away from the public eye and rarely gives interviews to talk about her past with Sierra On-Line.

Since aslan (" lion ") is a Turkish colloquial metaphor for a strong, courageous man, this gives the term a meaning close to " the milk of the strong.

Since metabolic conditions change rapidly, as a result of any of several factors, this is delayed news and gives no warning of a developing condition.