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 the map d restricts to 0 on I < sub > x </ sub >< sup > 2 </ sup > ( the reader should verify this ), d descends to a map from I < sub > x </ sub > / I < sub > x </ sub >< sup > 2 </ sup > to the dual of the tangent space, ( T < sub > x </ sub > M )< sup >*</ sup >.

Since the topology on R < nowiki ></ nowiki > X < nowiki ></ nowiki > is the ( X )- adic topology and R < nowiki ></ nowiki > X < nowiki ></ nowiki > is complete, we can in particular apply power series to other power series, provided that the arguments don't have constant coefficients ( so that they belong to the ideal ( X )): f ( 0 ), f ( X < sup > 2 </ sup >− X ) and f ( ( 1 − X )< sup >− 1 </ sup > − 1 ) are all well defined for any formal power series f ∈ R < nowiki ></ nowiki > X < nowiki ></ nowiki >.

Since an even number can be expressed as 2x, ( 2x )< sup > 2 </ sup >

Since (− 4 )< sup > 2 </ sup > − 4 ( 8 )

Since the product of an element in I with an element in R is in I ( as I is an ideal ), this product expression will be in I, and then ( a + b )< sup > n + m − 1 </ sup > is in I, therefore a + b is in the radical of I.

Since the factor ( z − a )< sup > n </ sup > does not depend on the variable of integration w, it can be pulled out:

Since Re ( CO )< sub > 5 </ sub > is isolobal with CH < sub > 3 </ sub >, < sup >+</ sup > is isolobal with CH < sub > 3 </ sub >< sup >+</ sup >.

Since – I acts ( – 1 )< sup > k </ sup > on the principal series and trivially on the remainder, these will give all the irreducible unitary representations of the Lorentz group, provided k is taken to be even.

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 mass numbers of these are 12, 13 and 14 respectively, the three isotopes of carbon are known as carbon-12, carbon-13, and carbon-14, often abbreviated to < sup > 12 </ sup > C, < sup > 13 </ sup > C, and < sup > 14 </ sup > C.

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 6 has 2 factors ( 2 and 3 ) it should have 2 < sup > 2 </ sup > idempotents.

Since terms of linear equations cannot contain products of distinct or equal variables, nor any power ( other than 1 ) or other function of a variable, equations involving terms such as xy, x < sup > 2 </ sup >, y < sup > 1 / 3 </ sup >, and sin ( x ) are nonlinear.

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 this carrier enzyme ( Na < sup >+</ sup >- Ca < sup > 2 +</ sup > translocator ) uses the Na gradient generated by the Na < sup >+</ sup >- K < sup >+</ sup > pump to remove Ca < sup > 2 +</ sup > from the intracellular space, slowing down the Na < sup >+</ sup >- K < sup >+</ sup > pump results in a permanently elevated Ca < sup > 2 +</ sup > level in the muscle, which may be the mechanism of the long-term inotropic effect of cardiac glycosides such as digoxin.

Since the NADH has lost electrons, NAD < sup >+</ sup > regenerates and is again available for glycolysis.

Since bases are proton acceptors, the base receives a hydrogen ion from water, H < sub > 2 </ sub > O, and the remaining H < sup >+</ sup > concentration in the solution determines pH.

Since these forces ( which are applied to K < sup >+</ sup > ions ) are now the same strength and oriented in opposite directions, the system is now in equilibrium.

Since the permeabilities to both ions were set to be equal, the membrane potential will, at the end of the Na < sup >+</ sup >/ K < sup >+</ sup > tug-of-war, end up halfway between E < sub > Na </ sub > and E < sub > K </ sub >.

Since the luminal membrane is quite leaky to Na < sup >+</ sup >, this causes a high back leak of Na < sup >+</ sup > into the tubule.

Since the action S is real and can be split into S < sub >+</ sub > which only depends on φ on the positive half-space and S < sub >−</ sub > which only depends upon φ on the negative half-space and if S also happens to be invariant under the combined action of taking a reflection and complex conjugating all the fields, then the previous quantity has to be nonnegative.

Since C < sup > 4 +</ sup > would be very unlikely and the signal grew stronger in pure hydrogen gas, he correctly assigned the species as H < sub > 3 </ sub >< sup >+</ sup >.

Since there is a high level of H < sub > 2 </ sub > in these atmospheres, this radiation can produce a significant amount of H < sub > 3 </ sub >< sup >+</ sup >.

Since then, H < sub > 3 </ sub >< sup >+</ sup > has been detected in numerous other molecular cloud sightlines, such as AFGL 2136 ,< ref name =" McCall1999 "> Mon R2 IRS 3, GCS 3-2 ,< ref name =" mgoto "> GC IRS 3, and LkHα 101 .< ref name =" Brittain2004 ">

Since then, H < sub > 3 </ sub >< sup >+</ sup > has been detected in many other diffuse cloud sightlines, such as GCS 3-2, GC IRS 3, and ζ Persei .< ref name =" McCall2004 ">