Thus, single-digit hash values are frequently restricted to 80 % of the size of the table.

Thus, it would seem all we have to do is compute the hash value of the substring we're searching for, and then look for a substring with the same hash value.

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

Thus, the function of antioxidant systems is not to remove oxidants entirely, but instead to keep them at an optimum level.

Thus the Bauhaus style, also known as the International Style, was marked by the absence of ornamentation and by harmony between the function of an object or a building and its design.

Thus the function cannot access memory out-of-bounds.

Thus any non-constant entire function must have a singularity at the complex point at infinity, either a pole for a polynomial or an essential singularity for a transcendental entire function.

Thus, it was not until two centuries had passed that in 1872 Karl Weierstrass presented the first definition of a function with a graph that would today be considered fractal, having the non-intuitive property of being everywhere continuous but nowhere differentiable.

Thus, any periodic summation in the frequency domain can be represented by a Fourier series, whose coefficients are samples of a related continuous time function:

Thus the DTFT of the s sequence is also the Fourier transform of the modulated Dirac comb function .< ref group =" note "> We may also note that:

Thus we have the important result that when a discrete data sequence, s, is proportional to samples of an underlying continuous function, s ( t ), one can deduce something about the continuous Fourier transform, S ( ƒ ).

Thus, we would say that the function is decomposed into functions.

Thus, the gamma function must be undefined at those points ; it is a meromorphic function with simple poles at the nonpositive integers.

Thus, a function with a codomain is invertible if and only if it is both injective ( one-to-one ) and surjective ( onto ).

Thus, the statement that f is an idempotent unary operation on S is equivalent to the statement that f is an idempotent element with respect to the function composition operation ∘ on functions from S to S.

Thus, they capture the idea of function composition within a set.

Thus, every MUMPS command or function name could be abbreviated from one to three letters in length, e. g. Quit ( exit program ) as Q, $ P

Thus for map projections we must introduce the concept of point scale, which is a function of position, and strive to keep its variation within narrow bounds.

Thus, the logarithm function is an isomorphism from the group of positive real numbers under multiplication to the group of real numbers under addition, represented as a function:

Thus, the neurochemical alterations which induce psychosis in otherwise healthy people are indicative that the root of the symptoms altered thalamocortical and hippocamalcortical transmission, and corresponding layer 5 / thalamus axis function, altering the crucial ordered cortiocortical layer 3 / 1 and 2 / 3 transmission necessary for rational language self-expression.

Thus, modern automatic rifles overlap to an extent in design and function with machine guns.

Thus, there is freshness not only in the individual movements of the dance but in the shape of their continuity as well.

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 Af if and only if Af.

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

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 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, giving grades and promotions based only on an individual's performance relative to a small local group, as is common, may reduce cooperative behaviors in the group.

Thus the only member churches of the present Anglican Communion existing by the mid-18th century were the Church of England, its closely linked sister church, the Church of Ireland ( which also separated from Roman Catholicism under Henry VIII ) and the Scottish Episcopal Church which for parts of the 17th and 18th centuries was partially underground ( it was suspected of Jacobite sympathies ).

Thus far the only people who have been attested with a high level of genetic, historical, linguistic and cultural research to be the descendants of the ancient Mesopotamians are the Assyrian Christians of Iraq and its surrounding areas in north west Iran, north east Syria and south eastern Turkey.

Thus, intuitionists are ready to accept a statement of the form " P or Q " as true only if we can prove P or if we can prove Q:

Thus they add two equivalents of bromine whereas an alkene adds only one equivalent.

Thus, negotiation generally features only in games for three or more people.

Thus, if the score was 3 – 2 with the bases loaded in the bottom of the ninth, and the batter smacked an " over the fence home run ", the game would end at 4 – 3, with the batter only allowed a double, and the runners officially stopped on 2nd and 3rd ( since they were not needed to win the game ).

Thus, a PICMG backplane can provide any number and any mix of ISA, PCI, PCI-X, and PCI-e slots, limited only by the ability of the SBC to interface to and drive those slots.

Thus only the target might be affected adversely.

Thus, a player should use Blackwood only when he can ascertain that the partnership holds at least second-round controls in all suits ( kings or, if a suit fit is found, singletons ).

Thus, only the immediate purchaser could recover for a product defect, and if a part was built up out of parts from parts manufacturers, the ultimate buyer could not recover for injury caused by a defect in the part.

Thus the castrati came to supplant both boys ( whose voices broke after only a few years ) and falsettists ( whose voices were weaker and less reliable ) from the top line in such choirs.

Thus, covalent bonding does not necessarily require the two atoms be of the same elements, only that they be of comparable electronegativity.

Thus, only Division I wins ( whether FBS, FCS, or FCS non scholarship ) are considered for playoff selection.

Thus, it would be biochemically possible to sustain life in environments that are only periodically consistent with life as we know it.

Thus the era was indeed a golden age but only for a small but powerful and highly visible elite.

Thus, only the Logos incarnate suffered and died on the Cross and therefore the Son was able to suffer without suffering.

Thus e. g. an O atom with only one Si bond ( a dangling bond ) can be considered a defect in silica.