Thus the Af are projections which correspond to some direct-sum decomposition of the space V.

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 his persistence values for some stem frequencies per meaning are: stem identical in 8 languages, 100% ; ;

Thus some provinces contained tribes which detested each other, and to them independence meant an opportunity for war.

Thus, while altruistic persons may under some circumstances be outcompeted by less altruistic persons at the individual level, according to group selection theory the opposite may occur at the group level where groups consisting of the more altruistic persons may outcompete groups consisting of the less altruistic persons.

Thus, although some things may be certain, they have little to do with Dasein's sense of care and existential anxiety, e. g., in the face of death.

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.

Thus, over the past 1, 500 years, some Christians have used the term Arian to refer to those groups that see themselves as worshiping Jesus Christ or respecting his teachings, but do not hold to the Nicene creed.

Thus while Persian-speakers were concentrated on the lower rungs of the work force, managers tended to be brought in from some distance.

Thus, during the abdication crisis of 1936, caused by Edward VIII's desire to marry Wallis Simpson, the consent of all realms concerned, along with, in some cases, new acts of parliament, was required in Britain and throughout the British Dominions to allow for Edward's stepping aside and to ensure that if he had any children they would have no claim to the thrones.

Thus aesthetic judgments might be seen to be based on the senses, emotions, intellectual opinions, will, desires, culture, preferences, values, subconscious behavior, conscious decision, training, instinct, sociological institutions, or some complex combination of these, depending on exactly which theory one employs.

For the case of an object that is small compared with the radial distance to its axis of rotation, such as a tin can swinging from a long string or a planet orbiting in a circle around the Sun, the angular momentum can be expressed as its linear momentum,, crossed by its position from the origin, r. Thus, the angular momentum L of a particle with respect to some point of origin is

Thus one finds brass instruments made of wood, like the alphorn, the cornett, the serpent and the didgeridoo, while some woodwind instruments are made of brass, like the saxophone.

Thus fines and noncustodial sentences may address the crimes seen as least serious, with lengthy imprisonment or ( in some jurisdictions ) capital punishment reserved for the most serious.

Thus if one chooses an infinite number of points in the closed unit interval, some of those points must get arbitrarily close to some real number in that space.

Thus literary criticism became highly theoretical and some of those practicing it began referring to the theoretical dimension of their work as " critical theory "-a philosophically inspired theory of literary criticism.

Thus miniatures were no longer required for game play, although some players continued to use them as a visual reference.

Thus, though the former understanding certainly enjoys the greatest popularity, the identification of Plotinus ’ opponents as Gnostic is not without some contention.

Thus, while D-Cinema is a defined standard, though one that is still partly being framed by SMPTE as of 2007, E-Cinema may be anything, ranging from a DVD player connected to a consumer projector to something that approaches the quality of D-Cinema without conforming to some of the standards.

Thus in some cases a single entity can essentially maintain controlling interest with only 33. 4 % of the outstanding shares.

Thus, some individuals and institutions will enter into a derivative contract to speculate on the value of the underlying asset, betting that the party seeking insurance will be wrong about the future value of the underlying asset.

Thus we are compelled to face the urbanization of the South -- an urbanization which, despite its dramatic and overwhelming effects upon the Southern culture, has been utterly ignored by the bulk of Southern writers.

Thus, paradoxically, the beat writers resort to `` religious '' metaphors: they are in search of mana, the spiritual, the numinous, but not anything connected with formal religion.

Thus, if corporations are not to run away with us, they must become quasi-governmental institutions, subject to public control and needs.

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 there are polynomials Af such that Af.

Thus pencils of tangents to Q are transformed into pencils of tangents.

Thus it follows that the secants of **zg are all invariant.

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 the first lines of one of Charles Wesley's hymns are as follows.

Thus in his friends are stranger than his sisters' strong stresses are normal for his and sisters', but in his friends are stranger than his sisters strong stresses are normal for friends and sisters.

Thus, the white middle and lower classes are arrayed against the Negro middle and lower classes.

Thus, during any given year ( A ) if the revenues from the residential service are $7,000,000, ( B ) if the operating expenses imputed to this class of service come to $5,000,000, and ( C ) if the net investment in ( or value of ) the plant and equipment deemed devoted to this service amounts to $30,000,000, the cost analyst will report that residential service, in the aggregate, has yielded a return of $2,000,000 or 6-2/3 per cent.

Thus, in the example cited above Fromm rests his whole case on the premise that the workers are being deprived unconsciously, unknowingly, of fulfillment, and then supports this with survey data reporting conscious, experienced frustrations.

Thus, the three main categories of antisubmarine warfare operations are defense of shipping, defense of naval forces, and area defense.

Thus, the most productive areas are not necessarily the most stressed.

Thus a number of operations of the order of Af are required.

Thus, if we are to observe American folklore in the twentieth century, we will do well to establish the relationships between folklore, nationalism and imperialism at the outset.

Thus, in `` The Story Of Ruth '' we have Ruth, Naomi, and Boaz and sets that are meticulously authentic.

" 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, one nucleus offers a more tightly bound position to an electron than does another nucleus, with the result that one atom may transfer an electron to the other.

Thus, is the result of applying the rule to.

Thus the adverb-adjective " effective " is used in a sense of " 1a: producing a decided, decisive, or desired effect ", and " capable of producing a result ".

Thus query optimization typically tries to approximate the optimum by comparing several common-sense alternatives to provide in a reasonable time a " good enough " plan which typically does not deviate much from the best possible result.

Thus query optimization typically tries to approximate the optimum by comparing several common-sense alternatives to provide in a reasonable time a " good enough " plan which typically does not deviate much from the best possible result.

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

Thus, a Good attribute is considered to be Great if you were to roll two plus sides, one minus side, and one blank — the minus side cancels out one of the plus sides and the remaining plus side raises the result by one step.

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, the monetary model predicts that the velocity of money will increase as a result of an excessive increase in the money supply.

Thus as a result of the three Carnatic Wars, the British East India Company gained exclusive control over the entire Carnatic region of India.

Thus, a too high or too low temperature will result in stellar instability.

Thus the formation of the Monty Python troupe has been referred to as a result of Cleese's desire to work with Palin and the chance circumstances that brought the other four members into the fold.

Thus, mediation is more likely to produce a result that is mutually agreeable for the parties.

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

Thus, weaker cases result in more lenient plea bargains, and stronger ones in relative harshness, but both result in an agreement.

Thus, for example, MOVEI A, 3 ( C ) adds 3 to the 18 lower bits of register C and puts the result in register A, without touching memory.

The rule is that “ n actor ’ s liability is limited to those physical harms that result from the risks that made the actor ’ s conduct tortious .” Thus, the operative question is " what were the particular risks that made an actor's conduct negligent?

Thus the solution is 2, not ½ as would be the result of an incorrect analysis.

Thus, measuring a quantum state described by complex coefficients ( a, b ,..., h ) gives the classical probability distribution and we say that the quantum state " collapses " to a classical state as a result of making the measurement.

Thus DA means that a cocking action separate from the trigger pull is unnecessary ; and every trigger pull will result in a complete cycle.

Thus, it is not enough to demonstrate that you have suffered the wrong in order to win a tort case, you must also have legally recognized damages that were directly or indirectly caused by the tortfeasor as a result of the tort, and be able to prove the extent of those damages.

Thus, although Buddhas possess no limitation from their side on their ability to help others, sentient beings continue to experience suffering as a result of the limitations of their own former negative actions.

Thus the secular power of the Bishop of Rome, i. e., the Pope, was reduced to one square mile, the smallest sovereign nation on earth ( as a result of the 1929 Lateran Treaty which established a Concordat between Vatican City and the nation of Italy ).

Thus, a few volts ' difference at the grid would make a large change in the plate current and could lead to a much larger voltage change at the plate ; the result was voltage and power amplification.