Thus at Mobile the annual mean is 67 ° F ( 19 ° C ), the mean for the summer 81 ° F ( 27 ° C ), and for the winter 52 ° F ( 11 ° C ); and at Valley Head, in De Kalb county, the annual mean is 59 ° F ( 15 ° C ), the mean for the summer 75 ° F ( 24 ° C ), and for the winter 41 ° F ( 5 ° C ).

Thus, all the antiderivatives of x < sup > 2 </ sup > can be obtained by changing the value of C in F ( x ) = ( x < sup > 3 </ sup >/ 3 ) + C ; where C is an arbitrary constant known as the constant of integration.

Thus, the divergence of the vector field F can be expressed as:

Thus, on 10 February of that year, a Junker 52 of the Réseau des Liaisons Aériennes Francaises ( R. L. A. F ), later known as Air France, landed in Plaisance.

Thus, as from 1945, the R. L. A. F operated the Paris-Mauritius line.

Thus given a set, a computable function has property F if and only if.

Thus, F (− q ) has the same magnitude as F ( q ) but they have the opposite phase, i. e., φ ( q ) = − φ ( q )

Thus a D whistle is fairly apt for playing both G and A, and a C instrument can be used fairly easily for F and G.

Thus, a battery designed to operate for at ( 77 ° F ) approximately would last only at ( 92 ° F ), and would last only at ( 107 ° F ).

Thus, a battery designed to operate for at ( 77 ° F ) would last at ( 113 ° F ).

Thus the tidal force can be linearized, resulting in the same formula for F < sub > T </ sub > as given above.

Thus D is isomorphic to the quotient group F / N.

Thus Spoke Zarathustra ( Nietzsche, F .) Penguin Classics: Penguin Publishing ( Originally published 1885 )

Thus one can define a ( unique ) unitary group of dimension n for the extension F < sub > q² </ sub >/ F < sub > q </ sub >, denoted either as U ( n, q ) or U ( n, q² ) depending on the author.

Thus a statement of the form is said to be true, under a particular interpretation, if there is some object in the domain of discourse of that interpretation that satisfies the predicate that the interpretation uses to assign meaning to the symbol Phil.

Thus the eigenfunction v satisfies

Thus we cannot get a simple generalization of the Schrödinger equation under the naive assumption that the wave function is a relativistic scalar, and the equation it satisfies, second order in time.

Thus a Dedekind domain is a domain which satisfies any one, and hence all four, of ( DD1 ) through ( DD4 ).

Given two tensor bundles E → M and F → M, a map A: Γ ( E ) → Γ ( F ) from the space of sections of E to sections of F can be considered itself as a tensor section of if and only if it satisfies A ( fs ,...) = fA ( s ,...) in each argument, where f is a smooth function on M. Thus a tensor is not only a linear map on the vector space of sections, but a C < sup >∞</ sup >( M )- linear map on the module of sections.

The notion is closely related to the idea of a multiplicative character in group theory, which is a homomorphism χ from a group G to the multiplicative group of a field F. Thus χ: G → F < sup >×</ sup > satisfies χ ( e ) = 1 ( where e is the identity element of G ) and

Thus, any estimator whose bias is given by a function satisfies

Thus, IST is an enrichment of ZFC: all axioms of ZFC are satisfied for all classical predicates, while the new unary predicate " standard " satisfies three additional axioms I, S, and T. In particular, suitable non-standard elements within the set of real numbers can be shown to have properties that correspond to the properties of infinitesimal and unlimited elements.

Thus the theory of interior algebras may be formulated using the closure operator instead of the interior operator, in which case one considers closure algebras of the form 〈 S, ·, +, ', 0, 1, < sup > C </ sup >〉, where 〈 S, ·, +, ', 0, 1 〉 is again a Boolean algebra and < sup > C </ sup > satisfies the above identities for the closure operator.

Thus only the fullness of the Beatific Vision satisfies this fundamental desire of the human soul to know God.

Thus the seeming contradiction is that a model which is itself countable, and which contains only countable sets, satisfies the first order sentence that intuitively states " there are uncountable sets ".

A simple calculation shows that this " some " S satisfies: for every x in the domain of T. Thus, S is the adjoint of T.

Thus there are polynomials Af such that Af.

Thus the Af are projections which correspond to some direct-sum decomposition of the space V.

Thus Af is divisible by the minimal polynomial P of T, i.e., Af divides Af.

Thus we see that Af and Af.

Thus, the study of the solutions to the equation Af is reduced to the study of the space of solutions of a differential equation of the form Af.

Thus Af if and only if Af.

Thus in the three-dice example Af, Af, and the independence assumption imply that the probability that the three dice fall ace, not-ace, ace in that order is Af.

Thus in independent voting, Af or 0.52 of the couples would agree.

Thus if E is sufficiently small, there can be only one intersection of C and Af near Q, for if there were more than one intersection for every E then the difference between C and Af near Q would not be a monotone function.

Thus Af is also continuous at Af, and in a neighborhood of Af which does not contain a tangent point.

Thus the multiplicity of Af for a given T must be an even number.

Thus, the efficiency **yt couplers is given by -- Af or approximately 50 percent each.

Thus, if corporations are not to run away with us, they must become quasi-governmental institutions, subject to public control and needs.

Thus, if public pressure sets the effective limit to the price that the industry may charge, this pressure is itself a function of the wage rate.

Thus, during any given year ( A ) if the revenues from the residential service are $7,000,000, ( B ) if the operating expenses imputed to this class of service come to $5,000,000, and ( C ) if the net investment in ( or value of ) the plant and equipment deemed devoted to this service amounts to $30,000,000, the cost analyst will report that residential service, in the aggregate, has yielded a return of $2,000,000 or 6-2/3 per cent.

Thus if the gyro and platform-controller combination maintains the platform with zero angular deviation about the **yf axis, the system can be rotated with an angular velocity Af if a torque is supplied to the gyro output axis Aj.

Thus, if one accelerometer is properly aligned, the other is also.

Thus, if we are to observe American folklore in the twentieth century, we will do well to establish the relationships between folklore, nationalism and imperialism at the outset.

Thus, red wine must, if possible, never be disturbed or shaken ; ;

Thus, if what is at issue is whether `` All S is P '', it is indifferent whether `` Some S is not P '' or `` No S is P '', since in either case the judgment in question is false.

Thus most Affidavits now require to be Apostilled if used for cross border issues.

For every group G there is a natural group homomorphism G → Aut ( G ) whose image is the group Inn ( G ) of inner automorphisms and whose kernel is the center of G. Thus, if G has trivial center it can be embedded into its own automorphism group.

Thus throwing a rock at someone for the purpose of hitting him is a battery if the rock in fact strikes the person and is an assault if the rock misses.

Thus, a court officer taking possession of goods under a court order may use force if reasonably necessary.

Thus, there is no assault if an ox gores a man.

Thus, if P is a program in L < sub > 2 </ sub > which is a minimal description of s, then InterpretLanguage ( P ) returns the string s. The length of this description of s is the sum of

Thus, if every historian were to claim that there was a solar eclipse in the year 1600, then though we might at first naively regard that as in violation of natural laws, we'd come to accept it as a fact.