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 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 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, despite the obvious analogy to planets revolving around the Sun, electrons cannot be described as solid particles.

Thus, small minority parties cannot easily enter the Bundestag and prevent the formation of stable majority governments as they could under the Weimar constitution.

Thus, the Big Bang theory cannot and does not provide any explanation for such an initial condition ; rather, it describes and explains the general evolution of the universe going forward from that point on.

Thus, for example, controversies in physics are limited to subject-areas where experiments cannot be carried out yet, whereas Benford's Law implies that controversy is inherent to politics, where communities must frequently decide on courses of action based on insufficient information.

Thus perceptions must already come parcelled into distinct " bundles " before they can be associated according to the relations of similarity and causality: in other words, the mind must already possess a unity that cannot be generated, or constituted, by these relations alone.

Thus we cannot define being, unity and similar concepts.

Thus, encountering something which evokes the implicit associations of an experience or sensation that cannot be remembered may lead to déjà vu.

Thus, the law of " interpenetrating opposites " records the inextricable interdependence of components: the " transformation of quantity to quality " defends a systems-based view of change that translates incremental inputs into alterations of state ; and the " negation of negation " describes the direction given to history because complex systems cannot revert exactly to previous states.

Thus, the meaning of sentences containing the word " good " cannot be explained entirely in terms of sentences not containing the word " good.

Thus cannot be too little, less than some constant.

Thus, as a simple instance posed by Hume, we cannot know with certainty by inductive reasoning that the sun will continue to rise in the East, but instead come to expect it to do so because it has repeatedly done so in the past.

Thus, higher hormone concentration alone cannot trigger the negative feedback mechanism.

Thus, one cannot expect to find a clear origin.

Thus the 32-bit integer and 32-bit floating-point objects can simply use the value directly ; whereas the 64-bit integer and 64-bit floating-point cannot use this method.

Thus, any real object cannot remain the same, also because of movement of its physical parts ( even much bigger than molecules ).

Thus, the main lesson from the argument is not that one risks big losses, but merely that one cannot always win.

If a statement P is provable, then it is certainly impossible to prove that there is no proof of P. But even if it can be shown that no disproof of P is possible, we cannot conclude from this absence that there is a proof of P. Thus P is a stronger statement than not-not-P.

Thus, the eventually attainable perfection and complexity of manufactured products, while they can be calculated in theory, cannot be predicted with confidence.

Thus, these cells cannot survive for long ' as they cannot exchange sufficient material to maintain active metabolism.

Thus, the statement " there are no contradictions in the Principia system " cannot be proven in the Principia system unless there are contradictions in the system ( in which case it can be proven both true and false ).

Thus, people cannot see or otherwise come into sensory contact with universals, but in order to conceive of universals, one must be able to conceive of these abstract forms.

Thus over-provisioning cannot be blindly assumed on the Internet.

Thus ' b ' and'd ' can rhyme ( both being ' voiced plosives '), as can ' bh ' and ' l ' ( which are both ' voiced continuants ') but ' l ', a ' voiced continuant ', cannot rhyme with ' ph ', a ' voiceless continuant '.