Thus, the rate of transitions between two stationary states is enhanced beyond that due to spontaneous emission.

Thus, the rate of transitions between two stationary states is enhanced beyond that due to spontaneous emission.

Thus the extremal ( shortest, longest or stationary ) paths contribute into this interference most as they can not be completely canceled out.

Thus, a small molecule that can penetrate every corner of the pore system of the stationary phase " sees " the entire pore volume and the interparticle volume, and will elute late ( when the pore-and interparticle volume has passed through the column ~ 80 % of the column volume ).

Thus, for m > 1, every cardinal that is either Π-indescribable or Σ-indescribable is both Π-indescribable and Σ-indescribable and the set of such cardinals below it is stationary.

Thus we see that the eigenstates of the Hamiltonian are stationary states, they only pick up an overall phase factor as they evolve with time.

Thus the wheels always see the same torque even when spinning at different speeds, including the case where one is stationary.

Thus a stationary point is always a critical point, but a critical point is not always a stationary point: it might also be a non-differentiable point.

Thus, while the inner ring is stationary, the outer ring is free to move with very little friction.

Thus, although the allied forces normally enjoy numerical superiority, typically 30 % of them are stationary each turn.

) Thus a static spacetime is a stationary spacetime satisfying this additional integrability condition.

Thus, the geometry of a stationary spacetime does not change in time.

Thus stationary distributions are related to eigenvectors for which the eigenvalue is unity.

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, knowing the students ' level on a developmental sequence provides information on the kind and level of knowledge they can assimilate, which, in turn, can be used as a frame for organizing the subject matter to be taught at different school grades.

Thus, measurement of the tension in the string identifies the inertial frame: it is the one where the tension in the string provides exactly the centripetal force demanded by the motion as it is observed in that frame, and not a different value.

Thus a ruler that is one meter long in one frame of reference will not be one meter long in a reference frame that is travelling at a velocity relative to the first frame.

Thus, the coordinates differences ( Δx, Δy, Δz ) and ( Δξ, Δη, Δζ ) are not intrinsic to the rod, but merely reflect the reference frame used to describe it ; by contrast, the length L is an intrinsic property of the rod.

Thus, the reading is given by the number of whole divisions that are visible on the scale of the frame, multiplied by 25 ( the number of thousandths of an inch that each division represents ), plus the number of that division on the thimble which coincides with the axial zero line on the frame.

Thus, when analyzing earthbound motions, the Earth frame is not an inertial frame, but rather rotates about the local vertical at an effective rate of radians per day.

Thus, during the early 1960s, Pontiacs featured the controversial " X " frame used on the big Chevys, as well as the complete Chevy lineup of OHV straight Sixes, small-block 283 and 327 cubic inch V8s, and the big-block 348 and 409 V8s.

Thus, when the frame story required the use of a mechanical device, Doc would merely find yet another of Uncle George's machines.

Thus the NTSC color subcarrier ended up as 3. 57954545 MHz ( actually 315 / 88 MHz ), the line scan frequency as 15734. 27 Hz and the frame rate 29. 97 Hz ( exactly 30 / 1. 001 Hz ).

Thus any black hole which has been isolated for a significant period of time can be described by the Kerr – Newman metric in an appropriately chosen reference frame.

Thus to an observer, a clock on the rocket moves slower, as do the clocks in any reference frame that is not comoving with them.

Thus P is a smooth manifold with a smooth map π: P → M which looks locally like the trivial bundle M × H → M. The frame bundle of M is a principal GL ( n )- bundle, while if M is a Riemannian manifold, then the orthonormal frame bundle is a principal O ( n )- bundle.

Thus, for example, in the center of mass frame, the total energy of an object or system is equal to its rest mass times c², a useful equality.

Thus the kicker may play the ball a second time only if the ball rebounds off the keeper, not if the ball rebounds directly off the goal frame without having touched another player.

" Thus, invariant mass is a natural unit of mass used for systems which are being viewed from their center of momentum frame ( COM frame ), as when any closed system ( for example a bottle of hot gas ) is weighed, which requires that the measurement be taken in the center of momentum frame where the system has no net momentum.

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

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, for aqueous media, we can think of the idealized organic active as an oleophilic or hydrophobic surface-active agent, and of an idealized builder as a oleophobic or hydrophilic surface-active agent.

Thus, the larvae of the intruder can develop at the expense of the rightful inhabitants and the store of beebread.

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, one can think of a dictionary entry as a word rather than a form.

Thus the unstressed it of it rarely snows here gets its significance from its use with snows: nothing can snow snow but `` it ''.

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, besides the training provided to youth in school, the existence of the school program can have supplementary benefits to industry which make it an asset to industrial development efforts.

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 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, man can readily deduce that the primary objective end of the conjugal act is procreation, the propagation of the race.

Thus, while it remains possible that the Babylonians and/or the Pythagoreans may perhaps have had the magic square of three before the Chinese did, more definite evidence will have to turn up from the Middle East or the Classical World before China can lose her claim to the earliest known magic square by more than a thousand years.

" Thus Boolos and Jeffrey are saying that an algorithm implies instructions for a process that " creates " output integers from an arbitrary " input " integer or integers that, in theory, can be chosen from 0 to infinity.

Thus an algorithm can be an algebraic equation such as y

Thus, an algorithm can be considered to be any sequence of operations that can be simulated by a Turing-complete system.

Thus although most species in the order are herbaceous, some no more than 15 cm high, there are a number of climbers ( e. g. some species of Asparagus ), as well as several genera forming trees ( e. g. Agave, Cordyline, Yucca, Dracaena ), some of which can exceed 10 m in height.

For every group G there is a natural group homomorphism G → Aut ( G ) whose image is the group Inn ( G ) of inner automorphisms and whose kernel is the center of G. Thus, if G has trivial center it can be embedded into its own automorphism group.

Thus areas can be measured in square metres ( m < sup > 2 </ sup >), square centimetres ( cm < sup > 2 </ sup >), square millimetres ( mm < sup > 2 </ sup >), square kilometres ( km < sup > 2 </ sup >), square feet ( ft < sup > 2 </ sup >), square yards ( yd < sup > 2 </ sup >), square miles ( mi < sup > 2 </ sup >), and so forth.

The evaluation order does not affect the value of such expressions, and it can be shown that the same holds for expressions containing any number of operations .< ref > Thus, when is associative, the evaluation order can be left unspecified without causing ambiguity, by omitting the parentheses and writing simply: