Thus there are polynomials Af such that Af.

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

Thus Af is divisible by the minimal polynomial P of T, i.e., Af divides Af.

Thus we see that Af and Af.

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 Af if and only if Af.

Thus F satisfies Af if and only if F has the form Af.

Thus in the three-dice example Af, Af, and the independence assumption imply that the probability that the three dice fall ace, not-ace, ace in that order is Af.

Thus in independent voting, Af or 0.52 of the couples would agree.

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 Af is also continuous at Af, and in a neighborhood of Af which does not contain a tangent point.

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

Thus, the efficiency **yt couplers is given by -- Af or approximately 50 percent each.

Thus far, the cases which have come before the courts have involved only the issue of referral where the job is vacant due to a strike -- condition ( 1 ) in the Regulation of the Secretary.

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, the planetary model of the atom was discarded in favor of one that described atomic orbital zones around the nucleus where a given electron is most likely to be observed.

" Thus Ambrose refused to be drawn into a false conflict over which particular local church had the " right " liturgical form where there was no substantial problem.

Thus, all the antiderivatives of x < sup > 2 </ sup > can be obtained by changing the value of C in F ( x ) = ( x < sup > 3 </ sup >/ 3 ) + C ; where C is an arbitrary constant known as the constant of integration.

Thus, current expected inflation reflects a weighted average of all past inflation, where the weights get smaller and smaller as we move further in the past.

Thus, many users of a given BBS usually lived in the same area, and activities such as BBS Meets or Get Togethers were common, where users of the board would gather at a local restaurant, the SysOp ’ s home or similar venue and meet face to face.

Thus the discovery report by the Berkeley group reads: " It is suggested that element 97 be given the name berkelium ( symbol Bk ) after the city of Berkeley in a manner similar to that used in naming its chemical homologue terbium ( atomic number 65 ) whose name was derived from the town of Ytterby, Sweden, where the rare earth minerals were first found.

In his fictional historical essay " The Hyborian Age ", Howard describes how the people of Atlantis — the land where his character King Kull originated — had to move east after a great cataclysm changed the face of the world and sank their island, settling where Ireland and Scotland would eventually be located, Thus they are ( in Howard's work ) the ancestors of the Irish and Scottish ( the Celtic Gaels ) and not the Picts, the other ancestor of modern Scots who also appear in Howard's work.

Thus, many understood Eutyches to be advocating Docetism, a sort of reversal of Arianism — where Arius had denied the consubstantial divinity of Jesus, Eutyches seemed to be denying his human nature.

Thus was born a new type of beef cattle for Central Brazil, with the name CANCHIM, derived from the name of a tree very common in the region where the breed was developed.

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, 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 Zheng became Qin Shi Huang, abolishing the system where the huang / di titles were reserved to dead and / or mythological rulers.

Thus in each term ( endothermic & exothermic ) the prefix refers to where heat goes as the reaction occurs.

Thus, if is the total number of trials and is the number of trials where the event occurred, the probability of the event occurring will be approximated by the relative frequency as follows:

Thus it has been proposed that Syrah red wine is named after Shiraz, a city in Persia where the grape was used to make Shirazi wine.

Thus, the debate of when, where, why and whether to use horseshoes is a hot topic today.

Thus for example Theocritus presents catalogues of heroines in two of his bucolic poems ( 3. 40 – 51 and 20. 34 – 41 ), where both passages are recited in character by lovelorn rustics.

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

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, an ionic bond is considered a bond where the ionic character is greater than the covalent character.

: Thus the profit maximization model predicts something about the effect of taxation on output, namely that output decreases with increased taxation.

Thus, there is freshness not only in the individual movements of the dance but in the shape of their continuity as well.

Thus jazz is transmuted into something holy, the sacred road to integration of being.

Thus the fictional detective is much more than a simple businessman.

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

Thus in both types attention is focused on the community itself, and its phenomenological life.

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

Thus human perception and human volition is the immanent cause of all social change and this most truly when the change reaches the civilizational level.

Thus, circular motion is itself one of the essential characteristics of completely perfect celestial existence.

Thus, in no ordinary sense of ' simplicity ' is the Ptolemaic theory simpler than the Copernican.

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 science is the savior of mankind, and in this respect Childhood's End only blueprints in greater detail the vision of the future which, though not always so directly stated, has nevertheless been present in the minds of most science-fiction writers.

Thus the copywriter in the world of the space merchants is the person who in earlier ages might have been a lyric poet, the person `` capable of putting together words that stir and move and sing ''.

Thus there is a clearer division of authority, administrative and legislative.

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, the need for the B-70 as a strategic weapon system is doubtful.

Thus technical efficiency is achieved at the expense of actual experience.

Thus, there is an added incentive to stay on the job.

Thus, the Span of its ossification was shortened and the center's ability to `` catch up '' in ossification is demonstrated.

Thus T is not diagonalizable.

Thus we do not score the number of bull's-eyes, and the random variable is not the number of successes.