" 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 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, 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, it is no mystical intuition, but an analyzable conception to say that man and his tradition can `` fall out of existence ''.

Thus, when you have prepared your foundation and laid the floor, these can be trucked to the site and erected with a small crew of friends in a weekend.

Thus the film can be applied to back-lighted translucent plastics faces ; ;

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, for aqueous media, we can think of the idealized organic active as an oleophilic or hydrophobic surface-active agent, and of an idealized builder as a oleophobic or hydrophilic surface-active agent.

Thus, the larvae of the intruder can develop at the expense of the rightful inhabitants and the store of beebread.

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, one can think of a dictionary entry as a word rather than a form.

Thus the unstressed it of it rarely snows here gets its significance from its use with snows: nothing can snow snow but `` it ''.

Thus in it's incredible what that boy can eat dominant stress is likely to be on incredible, and eat will have strong stress also.

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, for non-negative changes in the basic wage rate, the industry becomes the active wage-setter, since any increase in the basic wage rate can occur only by reason of industry acquiescence.

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 a man who is butting a stone wall at the office may become unusually aggressive in bed -- the one place he can still be champion.

Thus, man can readily deduce that the primary objective end of the conjugal act is procreation, the propagation of the race.

Thus, while it remains possible that the Babylonians and/or the Pythagoreans may perhaps have had the magic square of three before the Chinese did, more definite evidence will have to turn up from the Middle East or the Classical World before China can lose her claim to the earliest known magic square by more than a thousand years.

Thus although most species in the order are herbaceous, some no more than 15 cm high, there are a number of climbers ( e. g. some species of Asparagus ), as well as several genera forming trees ( e. g. Agave, Cordyline, Yucca, Dracaena ), some of which can exceed 10 m in height.

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 areas can be measured in square metres ( m < sup > 2 </ sup >), square centimetres ( cm < sup > 2 </ sup >), square millimetres ( mm < sup > 2 </ sup >), square kilometres ( km < sup > 2 </ sup >), square feet ( ft < sup > 2 </ sup >), square yards ( yd < sup > 2 </ sup >), square miles ( mi < sup > 2 </ sup >), and so forth.

The evaluation order does not affect the value of such expressions, and it can be shown that the same holds for expressions containing any number of operations .< ref > Thus, when is associative, the evaluation order can be left unspecified without causing ambiguity, by omitting the parentheses and writing simply: