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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
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Thus and let
Thus, he implicitly emphasized that it is God-made (" God has joined together "), " male and female ," lifelong (" let no one separate "), and monogamous (" a man ... his wife ").
Thus when God doth work, who shall let ( i. e. prevent ) it?
Thus, he cautiously states: " But knowing the sure number declared by Scripture, that is six hundred sixty and six, let them await, in the first place, the division of the kingdom into ten ; then, in the next place, when these kings are reigning, and beginning to set their affairs in order, and advance their kingdom, them learn to acknowledge that he who shall come claiming the kingdom for himself, and shall terrify those men of whom we have been speaking, have a name containing the aforesaid number, is truly the abomination of desolation.
Thus, writing to St. Jerome, St. Augustine said, " If that opinion of the creation of new souls is not opposed to this established article of faith let it be also mine ; if it is, let it not be thine.
Thus, this would let light of the right frequency to make this happen sum to a larger probability amplitude, and as such possess a larger probability of reaching the appropriate final point.
Thus, a person with severe coeliac disease is not allowed, let alone required, to eat any matzo other than gluten-free matzo.
Thus, if a censor uses this method to intercept messages, he or she may then let the letter be sent to the intended recipient who will be unaware that the secret message has already been intercepted by a third party.
Thus, there is no possibility of simply copying each list, let alone of using a template, as can be done, if different languages share one project with or without separate namespaces.
Thus let it be done!
Thus while the process is indeed deterministic, there is no better way to determine the being's will than to essentially run the experiment and let the being exercise it.
Thus Napoleon had to yet again disappoint the hopes of Italy, let Poland be crushed, and allow Prussia to triumph over Denmark in the Schleswig-Holstein question.
Thus, it is very unlikely that it will be possible to program a reasonably fast algorithm for playing the Go endgame flawlessly, let alone the whole Go game.
Thus by the time of King Sejong ( ruled 1418 – 1450 ), all branches of learning were rooted in this way of thinking although branches of Buddhism in Korea were still let to grow outside of the major political centres in a tolerance of other kinds of worship.
: Thus shall it preserve its integrity and self-respect ; and for the day of sickness let it be anointed with the clarified fat of the legs of a sacred leopard, and give it to drink a throstle's eggshell full of the juice of the custard apple in which has been dissolved three pinches of shredded rhinoceros horn, and apply it to piebald leeches.
Thus some elect to clean the interior with warm / hot soapy water and let it fully dry before re-filling.
Let L be the language of first-order arithmetic, and let N be the standard structure for L. Thus ( L, N ) is the " interpreted first-order language of arithmetic.
( Thus Christians are instructed to " let your light shine before men, that they may see your good works, and glorify your Father who is in heaven.
Thus Balon's hope that the Lannisters would let him rule as king in the Iron Islands if he betrayed the North comes to nothing, as Theon said it would.
* Book of Jeremiah: Thus saith the LORD, the God of Israel: I made a covenant with your fathers in the day that I brought them forth out of the land of Egypt, out of the house of bondage, saying: " At the end of seven years ye shall let go every man his brother that is a Hebrew, that hath been sold unto thee, and hath served thee six years, thou shalt let him go free from thee "; but your fathers hearkened not unto Me, neither inclined their ear.
Thus it is possible to have a genuine user with the correct knock let through the firewall even in the middle of a port attack from multiple IP's ( assuming the bandwidth of the firewall is not completely swamped ).
Thus and V
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, 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 and be
Thus the cocktail party would appear to be the ideal system, but there is one weakness.
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, 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, a child's Skeletal Age `` dots '' may be classified as `` advanced '' when they appear above the middle curve, `` moderate '' when they appear immediately above or below the middle curve, and `` delayed '' when they appear below the lower curve.
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 the multiplicity of Af for a given T must be an even number.
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, casework involving a limited number of interviews is still to be regarded in terms of the quality of service rendered rather than of the quantity of time expended.
Thus, this readiness to relax controls, evidenced in the Kohnstamm situation, appears to be a more general personality factor.
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, when more than one distinct form leads to a particular cell in the X-region, a chain of information cells must be created to accommodate the forms, one cell in the chain for each form.
Thus in a context in which there has been discussion of snow but mention of local conditions is new, dominant stress will probably be on here in it rarely snows here, but in a context in which there has been discussion of local weather but no mention of snow, dominant stress will probably be on snows.
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 while his theory or technique may not be oversubscribed, it is commonplace for bullish and bearish positions to become temporarily over-subscribed.
Thus one line in five from The Iliad and The Odyssey is to be found somewhere else in the two poems.
Thus, when specifically permitted, the operand of a given line on the Autocoder coding sheet may be continued in the operand of from one to four additional lines which immediately follow.
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, Sauternes or Barsacs should be very cold ; ;
Thus, red wine must, if possible, never be disturbed or shaken ; ;
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