Thus-boundedness-of-T-sup-∗-sup-on-its-do

Thus, boundedness of T ∗ on its domain does not imply boundedness of T. On the other hand, if T ∗ is defined on the whole space then T is bounded on its domain and therefore can be extended by continuity to a bounded operator on the whole space .< ref > Proof: being closed, the everywhere defined T ∗ is bounded, which implies boundedness of T ∗∗, the latter being the closure of T. See also for the case of everywhere defined T .</ ref > If the domain of T ∗ is dense, then it has its adjoint T ∗∗.

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Thus and T

Thus T is not diagonalizable.

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

Thus D' and N' commute with any polynomial in T ; ;

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

Thus for clocks on or near the geoid, T eph ( within 2 milliseconds ), but not so closely TCB, can be used as approximations to Terrestrial Time, and via the standard ephemerides T eph is in widespread use.

Thus this code number of a proof of contradition of T must be a non-standard number.

This identifies the tangent space T e at the identity with the space of left invariant vector fields, and therefore makes the tangent space at the identity into a Lie algebra, called the Lie algebra of G, usually denoted by a Fraktur Thus the Lie bracket on is given explicitly by = e .

Thus for a theory " T ", " G " is true, but not provable in " T ".

Thus, T ' ai chi ch ' uan in spite of its roots in martial arts has become similar in scope to Qigong, the purely meditative practice based on notions of circulation of qi.

Thus, one would expect to find a factor that roughly corresponds to " left " and " right ", as this is the dominant framing for politics in our society, but the basis of Eysenck's " tough / tender-minded " thesis ( the second, " T "- factor ) may well represent nothing beyond an abstract mathematical construct.

Thus, in DNA, the purines adenine ( A ) and guanine ( G ) pair up with the pyrimidines thymine ( T ) and cytosine ( C ), respectively.

Thus T: X → X ** is an injective linear mapping, though it is not surjective unless X is reflexive.

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 h = e < sup > D </ sup >, and formally inverting the exponential yields

Its change is equal to the amount of reversible work done on, or obtainable from, a system at constant T. Thus its appellation " work content ", and the designation A from Arbeit, the German word for work.

Thus the identifier " Model T " identifies the class ( model ) of automobiles that Ford's Model T comprises ; whereas the unique identifier " Model T Serial Number 159, 862 " identifies one specific member of that class — that is, one particular Model T car, owned by one specific person.

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.

There are 3 fundamental physical units in this equation: time t, mass m, and length l, and 4 dimensional variables, T, M, L, and g. Thus we need only 4 − 3 = 1 dimensionless parameter, denoted π, and the model can be re-expressed as

Thus T is uncountable: it cannot be placed in one-to-one correspondence with the set of natural numbers.

Thus the Fourier coefficients are just the values of the Fourier transform sampled on a grid of width 1 / T.

Thus, the definition of T 3 may cite T 0 , T 1 , or T 2½ instead of T 2 ( Hausdorffness ); all are equivalent in the context of regular spaces.

Thus and <

Thus areas can be measured in square metres ( m < sup > 2 </ sup >), square centimetres ( cm < sup > 2 </ sup >), square millimetres ( mm < sup > 2 </ sup >), square kilometres ( km < sup > 2 </ sup >), square feet ( ft < sup > 2 </ sup >), square yards ( yd < sup > 2 </ sup >), square miles ( mi < sup > 2 </ sup >), and so forth.

Thus, if P is a program in L 2 which is a minimal description of s, then InterpretLanguage ( P ) returns the string s. The length of this description of s is the sum of

Thus, if a two-dimensional array has rows and columns indexed from 1 to 10 and 1 to 20, respectively, then replacing B by B + c 1 -− 3 c 1 will cause them to be renumbered from 0 through 9 and 4 through 23, respectively.

Thus this class of molecules belong to the D 2d point group.

Thus, few drops of diamminesilver ( I ) hydroxide ( Ag ( NH 3 ) 2 OH ) reacts with terminal alkynes signaled by formation of a white precipitate of the silver acetylide.

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 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 single " rule ," like mapping every real number x to x < sup > 2 </ sup >, can lead to distinct functions and, depending on whether the images under that rule are understood to be reals or, more restrictively, non-negative reals.

Thus, oxygen is reduced to water and the original electron acceptors NAD < sup >+</ sup > and quinone are regenerated.

Thus, for example, the English form is is an apparent cognate of German ist, Latin est and Russian < nowiki > jest '</ nowiki >, even though the Germanic, Italic, and Slavic language groups split at least three thousand years ago.

Thus, the molecular formula of ethanol may be written in a paper in conformational, three-dimensional, full two-dimensional ( indicating every bond with no three-dimensional directions ), compressed two-dimensional ( CH 3 – CH 2 – OH ), separating the functional group from another part of the molecule ( C 2 H 5 OH ), or by its atomic constituents ( C 2 H 6 O ), according to what is discussed.

Thus ℵ 0 is a regular cardinal.

Thus and sup

Thus, if the Cl – ion is favored for reduction, then the water reaction is favored for oxidation producing gaseous oxygen, however experiments show gaseous chlorine is produced and not oxygen.

Thus and >∗</

cohomologous to c. Thus L defines a unique class in H < sup > 2 </ sup >( G, F < sup >∗</ sup >), which need not be trivial.

Thus on a manifold M of dimension n, a volume form is an n-form, a section of the line bundle Ω < sup > n </ sup >( M ) = Λ < sup > n </ sup >( T < sup >∗</ sup > M ), that is nowhere equal to zero.

Thus and >

The evaluation order does not affect the value of such expressions, and it can be shown that the same holds for expressions containing any number of operations .< ref > Thus, when is associative, the evaluation order can be left unspecified without causing ambiguity, by omitting the parentheses and writing simply:

Thus and on

Thus, on the highroad, a troop of soldiers `` marched in their own dust and sang '', while on the footpath one man walks alone.

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

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

Thus, an enemy would probably use this weapon for attack on static population centers such as large cities.

Thus, the client receives enough ego support to engage in constructive efforts on his own behalf.

Thus it is reasonable to believe that there is a significant difference between the two groups in their performance on this task after a brief `` structuring '' experience.

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 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 in Mary wrote an account of the trip first strong stress on Mary marks Mary as the first in a series of people who wrote accounts of the trip, strong stress on wrote marks the writing as the first of a series of actions of Mary's concerned with an account of her trip ( about which she may later have made speeches, for example ), and strong stress on trip makes the trip the first of a series of subjects about which Mary wrote accounts.

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, 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, when Dartmouth's Winter Carnival -- widely recognized as the greatest, wildest, roaringest college weekend anywhere, any time -- was broadcast over a national television hookup, Prexy John Sloan Dickey appeared on the screen in rugged winter garb, topped off by a tam-o'-shanter which he confessed had been acquired from a Smith girl.

Thus, moral issues concerning the nature of the legal and political processes take on theological dimensions.

Thus Baptist churches on the frontier took cognizance of charges against their members of drunkenness, fighting, malicious gossip, lying, cheating, sexual irregularities, gambling, horse racing, and failure to pay just debts.

Thus, the emperor could draw on sources not available to those with less puissant ancestors.

Thus there was really an excess of eighteenth-century charm as one of these light-weight pieces followed another on Saturday night.

Thus, giving grades and promotions based only on an individual's performance relative to a small local group, as is common, may reduce cooperative behaviors in the group.

Thus military expeditions employ anthropologists to discern strategic cultural footholds ; marketing professionals employ anthropology to determine propitious placement of advertising ; and humanitarian agencies depend on anthropological insights as means to fight poverty.

Thus, a Rabbi, a Catholic priest, and an agnostic might agree that, in this particular case, the best approach is to withhold extraordinary medical care, while disagreeing on the reasons that support their individual positions.

Thus agrarianism is not industrial farming, with its specialization on products and industrial scale.

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