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 an algorithm can be an algebraic equation such as y

Thus, for example, the same equation can be made to produce both a lyrical and melodic piece of music in the style of the mid-nineteenth century, and a fantastically dissonant cacophony more reminiscent of the avant-garde music of the 1950s and 1960s.

Thus, if n < sub > r </ sub > is the average number of times a new civilization reappears on the same planet where a previous civilization once has appeared and ended, then the total number of civilizations on such a planet would be ( 1 + n < sub > r </ sub >), which is the actual reappearance factor added to the equation.

Thus, the conservation of electric charge, as expressed by the continuity equation, gives the result:

Thus and, then the above equation becomes:

Thus, the fundamental solution may be found by performing the continued fraction expansion and testing each successive convergent until a solution to Pell's equation is found.

Thus the reflected ray has equation

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

Thus, this equation is sometimes known as the vector wave equation.

While this equation suggests that the activation energy is dependent on temperature, in regimes in which the Arrhenius equation is valid this is cancelled by the temperature dependence of k. Thus, E < sub > a </ sub > can be evaluated from the reaction rate coefficient at any temperature ( within the validity of the Arrhenius equation ).

Thus we cannot get a simple generalization of the Schrödinger equation under the naive assumption that the wave function is a relativistic scalar, and the equation it satisfies, second order in time.

Thus the Schrödinger equation may be seen as the far non-relativistic approximation of the Dirac equation when one may neglect spin and work only at low energies and velocities.

There are 3 fundamental physical units in this equation: time t, mass m, and length l, and 4 dimensional variables, T, M, L, and g. Thus we need only 4 − 3 = 1 dimensionless parameter, denoted π, and the model can be re-expressed as

Thus, the differential equation solutions can be classified into homotopy classes.

Thus, the system can be analyzed with Schrödinger's equation and the guiding equation, with an initial distribution for the particles in the system ( see the section on the conditional wave function of a subsystem for details ).

Thus, we have the guiding equation.

Thus, although the agenda of external assistance in the economic sphere are cumulative, and many of the policies suggested for nations in the earlier stages remain relevant, the basic purpose of American economic policy during the later stages of development should be to assure that movement into a stage of self-sustaining growth is not prevented by lack of foreign exchange.

Thus, Sauternes or Barsacs should be very cold ; ;

Thus the use of abeyance provided the security of a settlement for the pro-life campus club, while preserving the student society's voting membership's ability to take the matter back to court should they choose in the future to deny resources to the club.

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, in an agent-neutral theory, an actor's personal goals do not count any more than anyone else's goals in evaluating what action the actor should take.

Thus his calculations should have required an additional year, ending in 1967.

Thus, by 1939, both Church ( 1934 ) and Turing ( 1939 ), neither having knowledge of the other's efforts, had individually proposed that their " formal systems " should be definitions of " effective calculability "; neither framed their statements as theses.

In particular, one should think of the condition number as being ( very roughly ) the rate at which the solution, x, will change with respect to a change in b. Thus, if the condition number is large, even a small error in b may cause a large error in x.

Thus, Judaism rejects the notion that anyone can or should die for anyone else's sin.

Thus, when ClearType sacrifices color accuracy in order to increase luminance detail, the overall effect — as seen by human eyes — should be an improvement for most people.

Thus, the database's conceptual and external architectural levels should be maintained in the transformation.

Thus the halides DbX < sub > 5 </ sub > should react with water to form DbOX < sub > 3 </ sub >.

Thus, we should prioritize social reform over attempts to account for consequences, individual virtue or duty ( although these may be worthwhile attempts, provided social reform is provided for ).

Thus, there should be no need for prices.

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

Thus, the original owner should not be entitled to S2, but rather, some or all of S3.

Thus, while people have the right to private property, they should give it away as enlightened altruists.

Thus, according to Josephus, the traditional Jew was, should be, and can be, a loyal and peace-loving citizen.

Thus any individual who goes into a mountainous environment should have basic knowledge of knots and knot systems to increase safety and the ability to do interesting activities such as rappelling.

Thus the Syrian Mandate provided that the government should be based on an organic law which should take into account the rights, interests and wishes of all the inhabitants, and that measures should be enacted “ to facilitate the progressive development of Syria and the Lebanon as independent States ”.

Thus at any point there should be one special coordinate system, " at rest relative to the aether ".

" Thus, a practice that the police " should know is reasonably likely to evoke an incriminating response from a suspect ... amounts to interrogation ".

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

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, 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, a child's Skeletal Age `` dots '' may be classified as `` advanced '' when they appear above the middle curve, `` moderate '' when they appear immediately above or below the middle curve, and `` delayed '' when they appear below the lower curve.

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 the multiplicity of Af for a given T must be an even number.

Thus, casework involving a limited number of interviews is still to be regarded in terms of the quality of service rendered rather than of the quantity of time expended.

Thus, this readiness to relax controls, evidenced in the Kohnstamm situation, appears to be a more general personality factor.

Thus he may be referring to some concrete thing, or incident, in his immediate environment by some symbolic-sounding, hyperbolic reference to transcendental events on the global scene.

Thus, when more than one distinct form leads to a particular cell in the X-region, a chain of information cells must be created to accommodate the forms, one cell in the chain for each form.

Thus in a context in which there has been discussion of snow but mention of local conditions is new, dominant stress will probably be on here in it rarely snows here, but in a context in which there has been discussion of local weather but no mention of snow, dominant stress will probably be on snows.

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 while his theory or technique may not be oversubscribed, it is commonplace for bullish and bearish positions to become temporarily over-subscribed.

Thus one line in five from The Iliad and The Odyssey is to be found somewhere else in the two poems.

Thus, when specifically permitted, the operand of a given line on the Autocoder coding sheet may be continued in the operand of from one to four additional lines which immediately follow.

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, red wine must, if possible, never be disturbed or shaken ; ;