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 the possible values of f are well-ordered, this property can be established by proving for any r ( not in I ) with minimal value of f ( r ) in its class.

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 >.

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 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 surface area of a sphere of radius r is A = 4πr < sup > 2 </ sup >, then intensity I ( power per unit area ) of radiation at distance r is

Since the r < sup > th </ sup > failure comes last, it remains to choose the k trials with successes out of the remaining k + r − 1 trials.

Since the total number of trials is equal to the number of successes plus the number of failures, this definition differs from ours by adding constant r.

Since this * z changed to * r in the Scandinavian languages and in West Germanic ( German, Dutch, English, Frisian ), Verner's Law resulted in alternation of * s and * r in some inflectional paradigms, known as grammatischer Wechsel.

Since this time the basic Taijutsu of Kukishin-ryū is Takagi-ryū while the Bōjutsu, Spear and Naginata techniques have remained Kukishin-ryū. r

Since only the positions in space – time are being warped, not the fields, the Ψ are all zero and the X < sub > μ </ sub >< sup > ν </ sup > equal the Kronecker delta δ < sub > μ </ sub >< sup > ν </ sup >, where we have used μ instead of r for the index.

# Since r has been found, we can factor ( x − r ) out of p ( x ) repeatedly until we have removed all of the roots at r. Repeat this for any other multiple roots until there are no more multiple roots.

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

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 1990s, SO < sub > 2 </ sub > emissions have dropped 40 %, and according to the Pacific Research Institute, acid rain levels have dropped 65 % since 1976.

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 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 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 at this stress the slope of the material's stress-strain curve, E < sub > t </ sub > ( called the tangent modulus ), is smaller than that below the proportional limit, the critical load at inelastic buckling is reduced.

Since < tt > ns1 </ tt > is contained in < tt > example. org </ tt >, this requires resolving < tt > example. org </ tt > first, which presents a circular dependency.

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 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 1983 ,< ref >

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 the 2002 civil war in Côte d ' Ivoire, relations between it and Burkina Faso have been filled with accusations of Burkinabé support for rebels on one side and claims of mistreatment of Burkinabé workers on the other .< ref > Blaise Campaoré :<< La crise ivoirienne inquiète le Burkina >>, Le Figaro, December 11, 2005 </ ref > Côte d ' Ivoire remains Burkina Faso's largest regional trading partner in spite of their disputes and tens of thousands of Burkinabés continue to work in the Ivory Coast.

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 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 p ( x ) is irreducible, this means that p ( x ) = k ( x − a ), for some k ∈ F

Since 612 − 255 = 357 our next pair is

Since bosons carry one unit of angular momentum, the fermion's spin direction will flip from + to − ( or vice versa ) during such an exchange ( in units of the reduced Planck's constant ).

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 p is prime q − 1 must be a multiple of p. Of course, if the number m = < sup >( q − 1 )</ sup >⁄< sub > p </ sub > is odd, then q will be even, since it is equal to mp + 1.

Since ω = − dΦ / dt and k = + dΦ / dx, the wave velocity is v = dx / dt = ω / k.

Since 2000, a small amateur experimental planting has been underway in the Lake Champlain valley of Vermont at the Vermont Experimental Cold-Hardy Cactus Garden where winter temperatures can reach − 37 ° C with variable snowcover.

Proof: Since p is a prime number the only possible values of gcd ( p < sup > k </ sup >, m ) are 1, p, p < sup > 2 </ sup >, ..., p < sup > k </ sup >, and the only way for gcd ( p < sup > k </ sup >, m ) to not equal 1 is for m to be a multiple of p. The multiples of p that are less than or equal to p < sup > k </ sup > are p, 2p, 3p, ..., p < sup > k − 1 </ sup > p = p < sup > k </ sup >, and there are p < sup > k − 1 </ sup > of them.

Since methane clathrates are stable at a higher temperature than liquefied natural gas ( LNG ) (− 20 vs − 162 ° C ), there is some interest in converting natural gas into clathrates rather than liquifying it when transporting it by seagoing vessels.

Since the fundamental theorem of arithmetic applied to a non-zero integer n that is neither 1 nor − 1 also asserts uniqueness of the representation for p < sub > i </ sub > prime and e < sub > i </ sub > positive, a primary decomposition of ( n ) is essentially unique.

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 >.