Thus F satisfies Af if and only if F has the form Af.

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 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 if a represents Socrates then Phil ( a ) asserts the first proposition, p ; if a instead represents Plato then Phil ( a ) asserts the second proposition, q.

Thus there is a number x ≡ ( q − 1 )< sup >( p − 2 )</ sup > for which ( q − 1 )· x ≡ 1 ( mod p ), and therefore a number k for which ( q − 1 )· x − 1 = kp.

Thus 2 < sup >( q − 1 ) x </ sup >/ 2 < sup > kp </ sup >

Thus the initial assumption that p and q − 1 are relatively prime is untenable.

Thus, p / q is the root of qx − p = 0.

Thus for English the " e " box is the largest while the " j ", " k ", " q ", " x ", and " z " boxes are the smallest.

Thus the efficiency depends only on q < sub > C </ sub >/ q < sub > H </ sub >.

Thus, if one of the coordinates, say q < sub > k </ sub >, does not appear in the Lagrangian, the right-hand side of the equation is zero, and the left-hand side requires that

Thus, the absence of the ignorable coordinate q < sub > k </ sub > from the Lagrangian implies that the Lagrangian is unaffected by changes or transformations of q < sub > k </ sub >; the Lagrangian is invariant, and is said to exhibit a symmetry under such transformations.

Thus denoting the common side as a and the diagonals as p and q, in every rhombus

Thus, according to Euler's quadrilateral theorem, the circumradius can be expressed in terms of the diagonals p and q, and the distance x between the midpoints of the diagonals as

Thus, b < sub > q </ sub > is a symmetric bilinear form over K with matrix A. Conversely, any symmetric bilinear form b defines a quadratic form

Thus, this picture can be taken as a picture of eta as a function of q.

Thus the compound p → q is false if and only if both p is true and q is false.

Thus we are compelled to face the urbanization of the South -- an urbanization which, despite its dramatic and overwhelming effects upon the Southern culture, has been utterly ignored by the bulk of Southern writers.

Thus Faulkner reminds us, and wisely, that the `` new '' South has gradually evolved out of the Old South, and consequently its agrarian roots persist.

Thus the image of man has suffered complete fragmentation in personal and spiritual qualities, and complete objectification in sub-human and quasi-mechanistic powers.

Thus earth has fallen to the center of the universe.

Thus science is the savior of mankind, and in this respect Childhood's End only blueprints in greater detail the vision of the future which, though not always so directly stated, has nevertheless been present in the minds of most science-fiction writers.

Thus the government simultaneously undertook the vast burden of social security which had traditionally been privately discharged, and created a national scarcity which has engendered calamitous problems of social security.

Thus, there has come into being a situation in which the state must raise all of its own revenues and, in addition, must give assistance to its local governments.

Thus in a context in which there has been discussion of snow but mention of local conditions is new, dominant stress will probably be on here in it rarely snows here, but in a context in which there has been discussion of local weather but no mention of snow, dominant stress will probably be on snows.

Thus to has light stress both in that was the conclusion that I came to and in that was the conclusion I came to.

Thus, the public-limit price is raised further by a given wage increase the longer it has been since the previous price increase.

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, a well-planned adobe wall of the appropriate thickness is very effective at controlling inside temperature through the wide daily fluctuations typical of desert climates, a factor which has contributed to its longevity as a building material.

Thus the term asteroid has come increasingly to refer specifically to the small bodies of the inner Solar System out to the orbit of Jupiter, which are usually rocky or metallic.

Thus the negation of the axiom of choice states that there exists a set of nonempty sets which has no choice function.

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, at pH between 2. 2 and 9. 4, the predominant form adopted by α-amino acids contains a negative carboxylate and a positive α-ammonium group, as shown in structure ( 2 ) on the right, so has net zero charge.

Thus, if a two-dimensional array has rows and columns indexed from 1 to 10 and 1 to 20, respectively, then replacing B by B + c < sub > 1 </ sub >-− 3 c < sub > 1 </ sub > will cause them to be renumbered from 0 through 9 and 4 through 23, respectively.

In mathematics, an associative algebra A is an associative ring that has a compatible structure of a vector space over a certain field K or, more generally, of a module over a commutative ring R. Thus A is endowed with binary operations of addition and multiplication satisfying a number of axioms, including associativity of multiplication and distributivity, as well as compatible multiplication by the elements of the field K or the ring R.

Thus, study of this Barnum / Forer effect has been mostly focused on the level of acceptance of fake horoscopes and fake astrological personality profiles.

Thus, it has a year 0, the years before that are designated with negative numbers and the years after that are designated with positive numbers.

Thus the " Aegean Area " has now come to mean the Archipelago with Crete and Cyprus, the Hellenic peninsula with the Ionian islands, and Western Anatolia.

Thus, most recipes that use bay leaves will recommend their removal after the cooking process has finished.

Thus he has him murdered.