Thus 4, 000, 000, which has a logarithm ( in base 10 ) of 6. 602, has 7 as its nearest order of magnitude, because " nearest " implies rounding rather than truncation.

Thus the great-circle-arc model is similar to logarithm in that sums correspond to products.

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, the hash function only hints at the record's location — it tells where one should start looking for it.

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 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 jazz is transmuted into something holy, the sacred road to integration of being.

Thus the fictional detective is much more than a simple businessman.

Thus the cocktail party would appear to be the ideal system, but there is one weakness.

Thus in both types attention is focused on the community itself, and its phenomenological life.

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

Thus human perception and human volition is the immanent cause of all social change and this most truly when the change reaches the civilizational level.

Thus, circular motion is itself one of the essential characteristics of completely perfect celestial existence.

Thus, in no ordinary sense of ' simplicity ' is the Ptolemaic theory simpler than the Copernican.

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 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 copywriter in the world of the space merchants is the person who in earlier ages might have been a lyric poet, the person `` capable of putting together words that stir and move and sing ''.

Thus there is a clearer division of authority, administrative and legislative.

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, the need for the B-70 as a strategic weapon system is doubtful.

Thus technical efficiency is achieved at the expense of actual experience.

Thus, there is an added incentive to stay on the job.

Thus, the Span of its ossification was shortened and the center's ability to `` catch up '' in ossification is demonstrated.

Thus T is not diagonalizable.

Thus Af is divisible by the minimal polynomial P of T, i.e., Af divides Af.

Thus, the study of the solutions to the equation Af is reduced to the study of the space of solutions of a differential equation of the form Af.

Thus in the three-dice example Af, Af, and the independence assumption imply that the probability that the three dice fall ace, not-ace, ace in that order is Af.

Thus we do not score the number of bull's-eyes, and the random variable is not the number of successes.

Thus Af is also continuous at Af, and in a neighborhood of Af which does not contain a tangent point.