Since Af is ferromagnetic, we felt that any results obtained from the magnetically contaminated Af would be suspect.

Since Af are distinct prime polynomials, the polynomials Af are relatively prime ( Theorem 8, Chapter 4 ).

Since D' and N' commute with one another and Af, we see that D' and N' commute with T.

Since D and D' are both diagonalizable and they commute, they are simultaneously diagonalizable, and Af is diagonalizable.

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

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 the complex of singular lines is of order K and since there is no complex of invariant lines, it follows from the formula Af that the order of the involution is Af.

A final class of exceptional lines is identifiable from the following considerations: Since no two generators of Af can intersect, it follows that their image curves can have no free intersections.

Since two curves of symbol Af on Q intersect in Af points, it follows that there are Af lines of Af which are tangent to Aj.

Since the load L, under actual cutting conditions is caused by Af, it can be seen that Af

Since by the principle of optimality the remaining stages use an optimal Af-stage policy with respect to Af, we may enter section Af of the table at this state Af and read off the optimal policy for stage Af and the resulting state Af.

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.

Since P was never asserted as the only sufficient condition for Q, other factors could account for Q ( while P was false ).

** Hays, Samuel P. ' A History of Environmental Politics Since 1945 ( 2000 ), abridged version

Since the light is turned through 90 degrees by the reflection, the slope of the parabola at P must be 1.

Since the ideal windings have no impedance, they have no associated voltage drop, and so the voltages V < sub > P </ sub > and V < sub > S </ sub > measured at the terminals of the transformer, are equal to the corresponding EMFs.

Since Petermann v. International Brotherhood of Teamsters, 344 P. 2d 25 ( Cal. App.

Since E and P are defined separately, this equation can be used to define D. The physical interpretation of D is not as clear as E ( effectively the field applied to the material ) or P ( induced field due to the dipoles in the material ), but still serves as a convenient mathematical simplification, since Maxwell's equations can be simplified in terms of free charges and currents.

Since his plans for an early-morning coordinated attack were now infeasible, Lee instead ordered Longstreet to coordinate a massive assault on the center of the Union line, employing the division of George Pickett and brigades from A. P.

Every prime ideal P in a Boolean ring R is maximal: the quotient ring R / P is an integral domain and also a Boolean ring, so it is isomorphic to the field F < sub > 2 </ sub >, which shows the maximality of P. Since maximal ideals are always prime, prime ideals and maximal ideals coincide in Boolean rings.

Since there are other countries that are exporting the same widgets at a price of P < sub > world </ sub >, Japan's industry is threatened to go out of business should widgets be imported into their country without a tariff.

Since point P and O < sub > 2 </ sub > have only one velocity,

Since the point P ( and thus ) is arbitrary, it follows that

Since failure probabilities on fault trees tend to be small ( less than. 01 ), P ( A ∩ B ) usually becomes a very small error term, and the output of an OR gate may be conservatively approximated by using an assumption that the inputs are mutually exclusive events:

Since the power dissipation, P = I < sup > 2 </ sup > R, increases with the square of the current but only increases approximately linearly with the number of windings, the power lost in the windings can be minimized by reducing I and increasing the number of turns N proportionally.

Since furnaces had to be constantly burning, there were night shifts from 5: 00 P. M. to 3: 00 A. M.

Since the 1981 National Scout Jamboree, all Jamborees have been held at Fort A. P.

Since some indices, such as the S & P 500 and FTSE 100, are dominated by large company stocks, an index fund may have a high percentage of the fund concentrated in a few large companies.

Since HOU is used for William P. Hobby Airport, the new Houston-Intercontinental became IAH.

Since χ is a tensor, P is not necessarily colinear with E.

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 World War II the Oder-Neisse line divides this area, reducing the German part to the Uecker-Randow district.

Since the ending of apartheid, he has lived in Cape Town, and now divides his time between his global concert circuit, New York, and South Africa.

Since n divides p − 1, the Galois group has a cyclic subgroup H of order ( p − 1 )/ n.

Since a musical interval is a ratio of frequencies, the equal-tempered chromatic scale divides the octave ( which has a ratio of 2: 1 ) into twelve equal parts.

Since each field line generally begins at a north magnetic pole and ends at a south magnetic pole, the most general way of dividing simple flux systems involves four domains separated by two separatrices: one separatrix surface divides the flux into two bundles, each of which shares a south pole, and the other separatrix surface divides the flux into two bundles, each of which shares a north pole.

Since 1914, Cape Cod Bay has been connected to Buzzards Bay by the Cape Cod Canal, which divides the upper cape towns of Bourne and Sandwich.

:: Since is a prime, it divides one of the two factors.

Since p divides, p must divide.

Since retiring from pro football in 1994 because of injury, Mecklenburg divides his time between family, motivational speaking, and his ongoing charity involvements.

Since his retirement, Gustafson, divides his time between his home in southwest Austin, Texas and his ranch in Cameron, Texas, northeast of Austin.

Since is a common divisor of and, it also divides, in other words it divides its contents, which by induction ( since has fewer terms than ) is.