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

Thus, for example, V ← V + 1 is commonly understood to be a shorthand for ( V ,( V ) + 1 ).

Thus for an ion which traverses a 1 m flight path, across a time of 2000 ns, given an initial accelerating voltage of 5000 V and noting that one amu is 1 × 10 < sup >− 27 </ sup > kg, the mass-to-charge ratio ( more correctly the mass-to-ionisation value ratio ) becomes Z amu / charge.

Thus a point y in V is a cluster point of the net if and only if every neighborhood Y of y contains infinitely many elements of the sequence.

Thus K < sub > V </ sub > 1. 3 blockers could be potential immunosuppressants for the treatment of autoimmune disorders ( such as rheumatoid arthritis, inflammatory bowel disease and multiple sclerosis ).

Thus, Charles was the maternal uncle of Charles V of France, who solicited his relative's advice at Metz in 1356 during the Parisian Revolt.

The Battle of Pavia ( 1525 ) marks a watershed in the city's fortunes, since by that time, the former cleavage between the supporters of the Pope and those of the Holy Roman Emperor had shifted to one between a French party ( allied with the Pope ) and a party supporting the Emperor and King of Spain Charles V. Thus during the Valois-Habsburg Italian Wars, Pavia was naturally on the Imperial ( and Spanish ) side.

This question has two fundamental physical units: time t and length, and three dimensional variables: distance D, time taken T, and velocity V. Thus there is 3 − 2 = 1 dimensionless quantity.

Thus the total time is O ( E log E ) = O ( E log V ).

Thus, if V < sub > g </ sup > is the group velocity of a phonon wave packet, then the relaxation length is defined as:

Thus, suppose that ( x < sub > 1 </ sub >,..., x < sub > n </ sub >) is a choice of Cartesian coordinates, in terms of which the coordinates of the vector V are

Thus, G < sub > t </ sub >( V ) = exp ( tV ) is the exponential map of the vector tV.

) Thus the " saturation " voltage of a Darlington transistor is one V < sub > BE </ sub > ( about 0. 65 V in silicon ) higher than a single transistor saturation voltage, which is typically 0. 1-0. 2 V in silicon.

Thus the lands that make up the current province came under occupation of these, up to the barbarian invasions of the early V century A. D.

Thus if U < sub > p </ sub >, V < sub > p </ sub >, Y < sub > p </ sub > are three tangent vectors at p and a and b are real numbers, then

The Chinese words already introduced by his predecessors he did not alter, and in introducing new terms required in the translation of the Mahayana literature, the 大乘 Tasheng or greater development, he uses SH for SH and usually B for V. Thus the city Shravasti was in Pali Savatthi and in Chinese Sha-ba-ti.

Thus let V be a vector space of dimension n over the field of scalars S, and let each of f = ( X < sub > 1 </ sub >,..., X < sub > n </ sub >) and f ' = ( Y < sub > 1 </ sub >,..., Y < sub > n </ sub >) be a basis of V .< ref group =" note "> A basis f may here profitably be viewed as a linear isomorphism from R < sup > n </ sup > to V. Regarding f as a row vector whose entries are the elements of the basis, the associated linear isomorphism is then Also, let the change of basis from f to f ′ be given by

Thus, they were classified as a single species V. cholera: however, Hugh believed the differing features between the two could be of epidmiological importance, so El Tor vibrios were further classified as V. cholerae biotype eltor ( serogroup O1 ).

Thus, if what is at issue is whether `` All S is P '', it is indifferent whether `` Some S is not P '' or `` No S is P '', since in either case the judgment in question is false.

Thus a měizhōurén is an American in the generic sense, and a měiguórén is an American in the U. S. sense.

Thus, much of the spirits used in making a Canadian whisky, prior to aging, may have less grain-derived flavour than typical single malts or U. S. " straight " whiskeys.

Thus, it might be used to challenge the sharp distinction made in U. S. v. E. C. Knight ( 1895 ) between commerce, which was subject to federal regulation, and manufacturing, which was not.

Thus, in Wilson v. U. S. ( 1911 ), he asserted that corporate officers could not resist a subpoena for company records by invoking the Fifth Amendment's privilege against self-incrimination.

The media, in an attempt to explain the ideology of Ayatollah Khomeini and the Iranian Revolution to a Western audience described it as a " fundamentalist version of Islam " by way of analogy to the Christian fundamentalist movement in the U. S. Thus was born the term " Islamic fundamentalist ", which would come to be one of the most common usages of the term in the following years.

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 statement that f is an idempotent unary operation on S is equivalent to the statement that f is an idempotent element with respect to the function composition operation ∘ on functions from S to S.

Thus commercial lighting alone consumes about four to five million barrels per day ( equivalent ) of petroleum, in line with the alternate rationale above to estimate U. S. lighting energy consumption.

Thus enforcing that part of the treaty which was of interest to the United States, namely, " It granted the U. S. significant transit rights over the Panamanian isthmus "

Thus in four close U. S. elections ( 1824, 1876, 1888, and 2000 ), the candidate with the most popular votes still lost the election.

Thus far, Swiss authorities have blocked about 72 accounts totalling U. S .$ 22. 6 million.

Thus the area of S < sub > D </ sub > is obtained by integrating the length of the normal vector to the surface over the appropriate region D in the parametric uv plane.

Thus some of the suburbs of most of the developed world are comparable to several inner cities of the U. S. and Canada.

" Thus, tape drives on S / 360 were commonly addressed at 0C0-0C7.

Thus, for example, the Great Seal of the United States, among other uses, appears on the reverse of the one-dollar bill ; and several of the seals of the U. S. states appear on their respective state flags.

Thus, with the support of convention chairman Elihu Root, Taft's supporters outvoted Roosevelt's men, and the convention renominated incumbents William Howard Taft and James S. Sherman, making Sherman the first sitting vice-president to be nominated for re-election since John C. Calhoun in 1828.

CSMA / CA can optionally be supplemented by the exchange of a Request to Send ( RTS ) packet sent by the sender S, and a Clear to Send ( CTS ) packet sent by the intended receiver R. Thus alerting all nodes within range of the sender, receiver or both, to not transmit for the duration of the main transmission.

Thus, the population in the U. S. is not aging to the extent as those in Europe, Australia, or Canada.

The absolute number of cancer cases attributed to exposure to gasoline, however, is low, estimated a few cases per year in the U. S. Thus, the decision to require fewer aromatics has been criticized on the grounds of opportunity cost: the hundreds of millions of dollars spent on process redesign could have been spent on other, more fruitful ways of reducing deaths caused by cancer or automobiles.

Thus, Charles S. Maier suggests that a political economy approach: " interrogates economic doctrines to disclose their sociological and political premises .... in sum, regards economic ideas and behavior not as frameworks for analysis, but as beliefs and actions that must themselves be explained.

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 Af is divisible by the minimal polynomial P of T, i.e., Af divides 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 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 we do not score the number of bull's-eyes, and the random variable is not the number of successes.

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.