" 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, Euclid's algorithm, which computes the GCD of two integers, suffices to calculate the GCD of arbitrarily many integers.

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, 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, 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 wood from ancient structures can be matched to known chronologies ( a technique called cross-dating ) and the age of the wood determined precisely.

Thus generation has to be kept matched to the load, to prevent overloading failures of the generation equipment.

Thus on 1 May 1961 the TA divisional hedquarters were merged with regular army districts, which were matched with Civil Defence Regions to aid mobilisation for war.

Thus, the cone responses from two radiant spectra can be matched by appropriate choice of diagonal adaptation matrices D < sub > 1 </ sub > and D < sub > 2 </ sub >:

" Thus, despite his long proven corrupting influence and antagonism toward the state's native Yankee population, Curley managed to win over substantial numbers of them, won in a walk, and, once back in Congress, he compiled a voting record that matched his former opponent's in support of the Roosevelt administration's social agenda.

Thus the style of the entire series created an extremely eerie atmosphere, that remains rarely matched even by the best adult science-fiction on screen.

Thus, refusing to respond to health claims matched the emergent, masculine image of the New Marlboro.

Thus, our optimal matched filter is

Thus Moronobu's adoption of a grouping of 12 was conventional enough, particularly as such an arrangement afforded a context in which to alter the furnishings, clothing, and design patterns, matched more or less to the months of the year.

Thus, with the merged company's international subsidiaries continuing to use the Tomy name, it seemed appropriate to keep a domestic English name which matched that of the international subsidiaries.

Thus, in Chile, Argentina and the Canary Islands, godo was an ethnic slur used against European Spaniards, who in the early colony period often felt superior to the people born locally ( criollos ).

Thus locally compact spaces are as useful in p-adic analysis as in classical analysis.

Thus resources can often be used across a network as if they were available locally.

By Taylor's theorem, this is a locally free sheaf of modules with respect to the sheaf of germs of smooth functions of M. Thus it defines a vector bundle on M: the cotangent bundle.

Thus the brigand was seen as a local figure, locally supported, preying on strangers.

Thus traffic congestion is rare in the town and usually only experienced in the town centre locally known as The Village.

Thus, the VDCS in an independent entity managed locally by the milk producers and assisted by the District Milk Union.

Thus, it is most often the case that any solution to the optimal control problem is locally minimizing.

Thus a projective module is locally free.

Thus P is a smooth manifold with a smooth map π: P → M which looks locally like the trivial bundle M × H → M. The frame bundle of M is a principal GL ( n )- bundle, while if M is a Riemannian manifold, then the orthonormal frame bundle is a principal O ( n )- bundle.

Thus, regardless of any other aims attributed to it, the French intervention helped to stop the genocide locally and represents the only foreign intervention on the ground to have ended some of the killings after UNAMIR was reduced.

Thus classical concepts are to be understood as locally emergent in a relative-state sense and should no longer claim a fundamental role in the physical theory.

Thus, in the lead-up to Federation, the currency used in the Australian colonies consisted of British silver and copper coins, Australian minted gold sovereigns ( worth 1 pound ) and half sovereigns, locally minted copper trade tokens ( suppressed in 1881, some state earlier ( Pitt 2000, pp. 10 – 11 )) and private bank notes.

Thus, it has seen an increase in original, locally geared news stories and spot news photographs.

Thus, the challenge of real-time collaborative editing is to figure out exactly how to apply edits from remote users, which were originally created in versions of the document that never existed locally, and which may conflict with the user's own local edits.

Thus in 1540, when the Papal forces of Pope Paul III ousted the Baglioni family from Perugia in the brief war over salt taxes locally called the " Salt War " ( Guerra del Sale ), Deruta sided with the papacy against Perugia, an alliance that gained it a reduction in taxes.