Since the analysis must typically go beyond two-loop perturbation theory, the definition of the running coupling α < sub > TC </ sub >( μ ), it ’ s fixed point value α < sub > IR </ sub >, and the strength α < sub > χ SB </ sub > necessary for chiral symmetry breaking depend on the particular renormalization scheme adopted.

Since a Dirichlet character χ is completely multiplicative, its L-function can also be written as an Euler product in the half-plane of absolute convergence:

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.

Since the energy-momentum tensor T < sub > μν </ sub > is normally understood to be due to concentrations of matter in four-dimensional space, the above result is interpreted as saying that four-dimensional matter is induced from geometry in five-dimensional space.

However, the more difficult proof of the converse ( see below ) makes use of the vector space structure: Since both of the factors are vector spaces over Q, the tensor product can be taken over Q.

Since the trace of any tensor is independent of any coordinate system, the most complete coordinate-free decomposition of a symmetric tensor is to represent it as the sum of a constant tensor and a traceless symmetric tensor.

Since the tensor product concept is independent of any choice of basis, taking the tensor product of two vector bundles on M is routine.

Since A and B may both be regarded as R-modules, we may form their tensor product

Since the covariant derivative of a tensor field at a point depends only on value of the vector field at one can define the covariant derivative along a smooth curve in a manifold:

Since the dual of the pseudoscalar is the product of two " pseudo-quantities " it can be seen that the resulting tensor is a true tensor, and does not change sign upon an inversion of axes.

( Since cars have reasonable symmetrical shapes, the off-diagonal terms of the angular inertia tensor can usually be ignored.

Since the Riemann curvature depends only on the Levi-Civita connection, the variation of the Riemann tensor can be calculated as,

Since electromagnetic fields contribute only a traceless term to the stress-energy tensor,

Since giving the Einstein tensor does not fully determine the Riemann tensor, but leaves the Weyl tensor unspecified ( see the Ricci decomposition ), the Einstein equation may be considered a kind of compatibility condition: the spacetime geometry must be consistent with the amount and motion of any matter or nongravitational fields, in the sense that the immediate presence " here and now " of nongravitational energy-momentum causes a proportional amount of Ricci curvature " here and now ".

Since the mixed Kronecker delta is equivalent to the mixed metric tensor,

Since Hilbert spaces have inner products, one would like to introduce an inner product, and therefore a topology, on the tensor product that arise naturally from those of the factors.

Since Af and P divides Af for Af, we have 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.