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Thus and Euclid's
Thus the division form of Euclid's algorithm starts with a pair of positive integers and forms a new pair that consists of the smaller number and the remainder obtained by dividing the larger number by the smaller number.
Thus, in view of the ancillary relation in which Pappus's lemmas generally stand to the works to which they refer, it seems incredible that the first seven out of thirty-eight lemmas should be really equivalent ( as Chasles makes them ) to Euclid's first seven Porisms.

Thus and algorithm
" Thus Boolos and Jeffrey are saying that an algorithm implies instructions for a process that " creates " output integers from an arbitrary " input " integer or integers that, in theory, can be chosen from 0 to infinity.
Thus an algorithm can be an algebraic equation such as y
Thus, an algorithm can be considered to be any sequence of operations that can be simulated by a Turing-complete system.
Thus, asymmetric algorithm keys must be longer for equivalent resistance to attack than symmetric algorithm keys.
Thus, the algorithm can be written as a sequence of equations
) Thus far, no published FFT algorithm has achieved fewer than complex-number additions ( or their equivalent ) for power-of-two.
Thus, the algorithm matched locally unambiguous meanings ( for instance, while the word seal as such is ambiguous, there is only one meaning of " seal " in the animal subhierarchy ).
Thus, it is likely that the worst-case running time for any algorithm for the TSP increases exponentially with the number of cities.
Thus each timeslot sends and receives an 8-bit PCM sample, usually encoded according to A-law algorithm, 8000 times per second ( 8 x 8000 x 32 = 2, 048, 000 ).
Thus an algorithm to find isolated real roots of a low-degree polynomial in one variable may bear little resemblance to an algorithm for complex roots of a " black-box " function which is not even known to be differentiable.
Thus the general algorithm is saved in one place but the concrete steps may be changed by the subclasses.
Thus, for example, many library implementations of quicksort will switch to a simple loop-based insertion sort ( or similar ) algorithm once the number of items to be sorted is sufficiently small.
Thus, it is very unlikely that it will be possible to program a reasonably fast algorithm for playing the Go endgame flawlessly, let alone the whole Go game.
Thus, Bluestein's algorithm provides an O ( N log N ) way to compute prime-size DFTs, albeit several times slower than the Cooley – Tukey algorithm for composite sizes.
Thus, at each recursive stage, all of the polynomials are factorized into two parts of half the degree, each of which has at most three nonzero terms, leading to an O ( N log N ) algorithm for the FFT.
Thus, applying an edge detection algorithm to an image may significantly reduce the amount of data to be processed and may therefore filter out information that may be regarded as less relevant, while preserving the important structural properties of an image.
More specifically, the running time for testing whether H is a minor of G in this case is O ( n < sup > 3 </ sup >), where n is the number of vertices in G and the big O notation hides a constant that depends superexponentially on H. Thus, by applying the polynomial time algorithm for testing whether a given graph contains any of the forbidden minors, it is possible to recognize the members of any minor-closed family in polynomial time.
Thus any algorithm that can be expressed primarily as operations on arrays and matrices can run almost as quickly as the equivalent C code.
Thus the amount of time taken and the number of elementary operations performed by the algorithm differ by at most a constant factor.
Thus, the stages of the algorithm are iterated to ensure aggressive simplification and coalescing.

Thus and which
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 with regard to the loss of tradition, in the change from order to disorder the metaphysics of change works itself out as a disruption of the individual soul, a change in which man continues as an objective ontological existent, but no longer as a man.
Thus Burns's `` My love is like a red, red rose '' and Hopkins' `` The thunder-purple sea-beach, plumed purple of Thunder '' although clearly intelligible in content, hardly present ideas of the sort with which we are here concerned.
Thus Trevelyan repeats the story which pictured Victor Emmanuel as refusing to abandon the famous Statuto at the insistence of General Radetzky.
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, as a development program is being launched, commitments and obligations must be entered into in a given year which may exceed by twofold or threefold the expenditures to be made in that year.
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 the last few years, a number of programs which looked very promising at the time their development was commenced have since been completely eliminated.
Thus was invented the single thread sewing machine, which Mr. Gibbs in partnership with Mr. Willcox decided to bring to Brown & Sharpe with the proposal that the small company undertake its manufacture.
Thus, direct comparisons can be drawn with free burning arcs which have been studied in detail during the past years and decades by numerous investigators ( Ref. 3 ).
Thus the Af are projections which correspond to some direct-sum decomposition of the space V.
Thus Af is also continuous at Af, and in a neighborhood of Af which does not contain a tangent point.
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 in Mary wrote an account of the trip first strong stress on Mary marks Mary as the first in a series of people who wrote accounts of the trip, strong stress on wrote marks the writing as the first of a series of actions of Mary's concerned with an account of her trip ( about which she may later have made speeches, for example ), and strong stress on trip makes the trip the first of a series of subjects about which Mary wrote accounts.
Thus, besides the training provided to youth in school, the existence of the school program can have supplementary benefits to industry which make it an asset to industrial development efforts.
Thus far, the cases which have come before the courts have involved only the issue of referral where the job is vacant due to a strike -- condition ( 1 ) in the Regulation of the Secretary.
Thus, when specifically permitted, the operand of a given line on the Autocoder coding sheet may be continued in the operand of from one to four additional lines which immediately follow.
Thus, when Dartmouth's Winter Carnival -- widely recognized as the greatest, wildest, roaringest college weekend anywhere, any time -- was broadcast over a national television hookup, Prexy John Sloan Dickey appeared on the screen in rugged winter garb, topped off by a tam-o'-shanter which he confessed had been acquired from a Smith girl.
Thus some provinces contained tribes which detested each other, and to them independence meant an opportunity for war.
Thus the image of God is that which makes a man a man and not an oyster ; ;
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.

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