Thus, if people are accustomed to using a particular machine name to access a particular service, their access will break when the service is moved to a different machine.

Thus a third style, the " curved lip " style was produced, and all official and third-party cartridges during the console's lifespan were released ( or re-released ) using this style.

Thus, the Bayesian statistician needs either to use informed priors ( using relevant expertise or previous data ) or to choose among the competing methods for constructing " objective " priors.

Thus, computing includes designing and building hardware and software systems for a wide range of purposes ; processing, structuring, and managing various kinds of information ; doing scientific studies using computers ; making computer systems behave intelligently ; creating and using communications and entertainment media ; finding and gathering information relevant to any particular purpose, and so on.

Thus, what started as an effort to translate between languages evolved into an entire discipline devoted to understanding how to represent and process natural languages using computers.

Thus dates can be sorted using simple string comparison algorithms, and indeed by any left to right collation.

Thus the analysis using Newton's laws of motion can proceed as if the reference frame was inertial, provided the fictitious force terms are included in the sum of external forces.

Thus most database systems nowadays are software systems running on general-purpose hardware, using general-purpose computer data storage.

Thus, instead of using rules they no longer remember, as knowledge engineers suppose, the expert is forced to remember rules they no longer use.

Thus one must be careful with the notation when using these functions, because various reputable references and software packages use different conventions in the definitions of the elliptic functions.

In a digital implementation, the number of computations performed per sample is proportional to N. Thus the mathematical problem is to obtain the best approximation ( in some sense ) to the desired response using a smaller N, as we shall now illustrate.

Thus, the third phase of the war ( roughly August 1920 – July 1921 ) involved the IRA taking on a greatly expanded British force, moving away from attacking well defended barracks and instead using ambush tactics.

Thus, many typical household applications already in existence like word processing and email were designed specifically to avoid using floating point operations.

Thus many equipment manufacturers add on their own particular requirements or tighten the tolerance on a general specification to meet their particular needs ( or doing a different set of tests or using different / own testbed engine ).

Thus, the effect of using this procedure is to make activity equal to the numerical value of concentration.

Thus studies of different populations using the same contraceptive will yield different values for the index.

Thus, a list of possible rational roots of the equation can be derived using the formula.

Thus, the so-called blub paradox ( after a hypothetical programming language of average complexity called ' Blub ') says that anyone preferentially using some particular programming language will ' know ' that it is more powerful than some, but not that it is less powerful than others.

Thus using as defined above a Scheme implementation would rewrite "" as "", which reduces implementation's task to that of coding procedure instantiations.

Thus far, most DNA sequencing has been performed using the chain termination method developed by Frederick Sanger.

Expected to make use of devices with which listeners could press a button to indicate whether they liked or disliked a particular piece of music, Adorno bristled with distaste and astonishment: “ I reflected that culture was simply the condition that precluded a mentality that tried to measure it .” Thus Adorno suggested using individual interviews to determine listener reactions and, only three months after meeting Lasarzfeld, completed a 160-page memorandum on the Project ’ s topic, “ Music in Radio .” Adorno was primarily interested in how the musical material was affected by its distribution through the medium of radio and thought it imperative to understand how music was affected by its becoming part of daily life.

Thus, using Planck's constant h to determine the energy of the photons based upon their frequency, the energy of ejected electrons should also increase linearly with frequency ; the gradient of the line being Planck's constant.

Thus there is no simple way to find an equivalent in the proleptic Julian calendar of a date quoted using the Roman pre-Julian calendar ( AUC or by reference to consuls ).

Thus it is usually called Euler's phi function or simply the phi function.

Thus the incircle radius is no larger than half the circumcircle radius ( Euler's triangle inequality ).

Thus, if k and n are coprime, z < sup > k </ sup > is also a primitive nth root of unity, and therefore there are φ ( n ) ( where φ is Euler's totient function ) distinct primitive nth roots of unity.

Thus, according to Euler's quadrilateral theorem, the circumradius can be expressed in terms of the diagonals p and q, and the distance x between the midpoints of the diagonals as

Thus, while more computationally intensive than Euler's method, the midpoint method generally gives more accurate results.

By a theorem of Gelfand and Naimark, given a B * algebra A there exists a Hilbert space H and an isometric *- homomorphism from A into the algebra B ( H ) of all bounded linear operators on H. Thus every B * algebra is isometrically *- isomorphic to a C *- algebra.

Thus, what is known as the extreme value theorem in calculus generalizes to compact spaces.

Thus, in a sense, there is a different completeness theorem for each deductive system.

Thus the no-cloning theorem holds in full generality.

Thus if we wish to claim that no-cloning is a uniquely quantum result, some care is necessary in stating the theorem.

) Thus, the group of conformal transformations in spaces of dimension greater than 2 are much more restricted than the planar case, where the Riemann mapping theorem provides a large group of conformal transformations.

Thus no formal system ( satisfying the hypotheses of the theorem ) that aims to characterize the natural numbers can actually do so, as there will be true number-theoretical statements which that system cannot prove.

Thus, by the principal axis theorem, the second fundamental form is

Thus according to the central limit theorem, we can model jitter as a gaussian random variable.

Thus Tychonoff's theorem joins several other basic theorems ( e. g. that every nonzero vector space has a basis ) in being equivalent to AC.

Thus one can see that the Cayley – Hamilton theorem must be true.

Thus thermal radiation contains information about the body that emitted it, while Hawking radiation seems to contain no such information, and depends only on the mass, angular momentum, and charge of the black hole ( the no-hair theorem ).

Thus, non-real roots of real polynomials occur in complex conjugate pairs ( see Complex conjugate root theorem ).

Thus, proving the claim will prove the theorem.

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 whenever a theorem and its reciprocal are true we have a biconditional.

Thus we get the right side of Green's theorem

Thus the Picard – Lindelöf theorem guarantees the existence and uniqueness of the solution.

Thus all geodesics leaving a point will eventually reconverge after a finite time, provided the appropriate energy condition holds, a result also known as the focusing theorem.

Thus reflections generate the orthogonal group, and this result is known as the Cartan – Dieudonné theorem.

Thus h can be extended to an entire bounded function which by Liouville's theorem implies it is constant.

Thus the coimage of f is a quotient algebra of X much as the image of f is a subalgebra of Y ; and the bijection between them becomes an isomorphism in the algebraic sense as well ( this is the most general form of the first isomorphism theorem in algebra ).

Thus one obtains Stone's representation theorem for Boolean algebras.