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.

where D denotes the continuum derivative operator, mapping f to its derivative f. The expansion is valid when both sides act on analytic functions, for sufficiently small h. Thus, T < sub > h </ sub >= e < sup > D </ sup >, and formally inverting the exponential yields

Thus, the derivative of f ( x ) becomes for an infinitesimal, where st (·) denotes the standard part function, which associates to every finite hyperreal the unique real infinitely close to it.

Thus, velocity is the time rate of change of position, and the dot denotes the derivative with respect to time.

Thus, if its inputs are an array x of length N, we can imagine extending this array to ( x, – x < sub > R </ sub >, – x, x < sub > R </ sub >, ...) and so on, where x < sub > R </ sub > denotes x in reverse order.

: Thus, the MDCT of 2N inputs ( a, b, c, d ) is exactly equivalent to a DCT-IV of the N inputs: (– c < sub > R </ sub >– d, a – b < sub > R </ sub >), where R denotes reversal as above.

Thus Ólafs ríma Haraldssonar denotes an epic about Ólafr Haraldsson in one ríma, while Núma rímur are a multi-part epic on Numa Pompilius.

According to Monier-Williams ( 1964 ), the etymology of sukha is " said to be su + kha and to mean originally ' having a good axle-hole '...." Thus, for instance, in the Rig Veda sukha denotes " running swiftly or easily " ( applied, e. g., to chariots ).

Thus, there has come into being a situation in which the state must raise all of its own revenues and, in addition, must give assistance to its local governments.

Thus, he supported funding to the state to assist the expansion of the East Tennessee and Virginia Railroad.

Thus the necessary and sufficient conditions for the truth of a proposition of law simply involved internal logic and consistency, and that the state's agents used state power with responsibility.

Thus a description of the cat during the course of the experiment — having been entangled with the state of a subatomic particle — becomes a " blur " of " living and dead cat.

Thus, texts like the Annals will often state whether the calendar they use ( the calendar of Lu ) is in phase with the Royal calendar ( used by the Zhou kings ).

Thus, culture has replaced the state as the locus of war.

Thus dubnium is expected to form a stable + 5 state.

Thus, it is easier to state what God is not, than to state what God is:

Thus, the law of " interpenetrating opposites " records the inextricable interdependence of components: the " transformation of quantity to quality " defends a systems-based view of change that translates incremental inputs into alterations of state ; and the " negation of negation " describes the direction given to history because complex systems cannot revert exactly to previous states.

Thus, while a permit to fill non-federal wetlands might require a permit from a single state agency, larger and more complex endeavors — for example, the construction of a coal-fired power plant — might require approvals from numerous federal and state agencies.

Thus, if the rate of any pathway changes, both the excited state lifetime and the fluorescence quantum yield will be affected.

Thus, a piece of iron in its lowest energy state (" unmagnetized ") generally has little or no net magnetic field.

Thus in diplomatic affairs, heads of state are often the first person to greet an important foreign visitor.

Thus in March 1979 Colonel Muammar Gaddafi, who kept absolute power ( until his overthrow in 2011 referred to as " Guide of the Revolution "), after ten years as combined Head of State and Head of government of the Libyan Jamahiriya (" state of the masses "), styled Chairman of the Revolutionary Command Council, formally transferred both qualities to the General secretaries of the General People's Congress ( comparable to a Speaker ) respectively to a Prime Minister, in political reality both were his creatures.

Thus, the nearest formal oxidation state of iron in Hb-O < sub > 2 </ sub > is the + 3 state, with oxygen in the-1 state ( as superoxide. O < sub > 2 </ sub >< sup >-</ sup >).

Thus, even when the Electors of Saxony were Catholics during the eighteenth century, they continued to preside over the corpus evangelicorum, since the state of Saxony was officially Protestant.

Thus, the first East Slavic state, Kievan Rus ', emerged in the 9th century along the Dnieper River valley.

Thus, the state required service from both the old and the new nobility, primarily in the military.

Thus the argument also involved the issue of the proper relationship between church and state.

Thus, a short path between the hot surface and the cold surface is necessary, typically by suspending a hot plate covered with a film of feed next to a cold plate with a line of sight in between.

Thus stealing, for instance, might be considered to be always immoral, even if done to promote some other good ( e. g., stealing food to feed a starving family ), and even if it does in the end promote such a good.

Thus, parents who lack the financial means to feed their children cannot use necessity as a defense if they steal food.

Thus, for instance, every male among the Hovas, from the age of sixteen to sixty, had either to pay twenty-five francs a year, or give fifty days of labour of nine hours a day, for which he was to be paid twenty centimes, a sum sufficient to feed him.

Thus, the feed on relation among life forms is intransitive, in this sense.

Thus, a short path between the hot surface and the cold surface is necessary, typically by suspending a hot plate covered with a film of feed next to a cold plate with a line of sight in between.

Thus natural gas is one of the most wanted feed stocks for petrochemicals production.

The function first increases nearly exponentially with increasing temperature, then slowly increases to the maximum value of 1. 0 at a temperature of T, then declines steeply at higher temperatures .”( Cooke 471 ) Thus proving that sunfish are opportunistic feeders, but also that temperature plays a large role in how often they feed.

Thus, social solidarity is maintained in more complex societies through the interdependence of its component parts ( e. g., farmers produce the food to feed the factory workers who produce the tractors that allow the farmer to produce the food )

Thus, all water that falls at this point can flow to the Pacific, Atlantic, or Arctic oceans ( when Hudson Bay is considered an Arctic tributary ), making the locale ( and surrounding Waterton-Glacier International Peace Park ) one of the few places on the Earth whose waters feed three oceans.

Thus, Konishi Yukinaga, the commander of the contingent of troops in Pyongyang could not move further north due to lack of supplies, nor could more troops be sent to him because there was not enough food to feed them.