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Thus and unitary
Thus, once we settle on any unitary representation of the gammas, it is final provided we transform the spinor according the unitary transformation that corresponds to the given Lorentz transformation.
) Thus there is a unitary freedom in the ket mixture or ensemble that gives the same density operator.
Thus one can define a ( unique ) unitary group of dimension n for the extension F < sub > q² </ sub >/ F < sub > q </ sub >, denoted either as U ( n, q ) or U ( n,) depending on the author.
Thus, unitary operators are just automorphisms of Hilbert spaces, i. e., they preserve the structure ( in this case, the linear space structure, the inner product, and hence the topology ) of the space on which they act.
Thus we define the representations of G on an Hilbert space H to be those group homomorphisms, ρ, which arise from continuous actions of G on H. We say that a representation ρ is unitary if ρ ( g ) is a unitary operator for all g ∈ G ; i. e., for all v, w ∈ H. ( I. e.
Thus it is not actually known what the correct term for the areas governed by the new unitary councils is or even if there is one.
Thus, 60 is a unitary perfect number, because its unitary divisors, 1, 3, 4, 5, 12, 15 and 20 are its proper unitary divisors, and 1 + 3 + 4 + 5 + 12 + 15 + 20 = 60.
Thus module can be extended to mean more generally a unitary part that gives the measuring unit for the whole.
Thus he offers a unitary, coherent and complete vision of this system.

Thus and structure
Thus, at pH between 2. 2 and 9. 4, the predominant form adopted by α-amino acids contains a negative carboxylate and a positive α-ammonium group, as shown in structure ( 2 ) on the right, so has net zero charge.
In mathematics, an associative algebra A is an associative ring that has a compatible structure of a vector space over a certain field K or, more generally, of a module over a commutative ring R. Thus A is endowed with binary operations of addition and multiplication satisfying a number of axioms, including associativity of multiplication and distributivity, as well as compatible multiplication by the elements of the field K or the ring R.
Thus the record and array data structures are based on computing the addresses of data items with arithmetic operations ; while the linked data structures are based on storing addresses of data items within the structure itself.
Thus differential geometry may study differentiable manifolds equipped with a connection, a metric ( which may be Riemannian, pseudo-Riemannian, or Finsler ), a special sort of distribution ( such as a CR structure ), and so on.
Thus Esperanto achieved a stability of structure and grammar similar to that which natural languages enjoy by virtue of their native speakers and established bodies of literature.
Thus in the system of units in which the fine structure constant is fixed, the observational claim is that the speed of light is time-dependent.
Thus, a unique trajectory in the thunderstorm is sufficient to explain the layer-like structure of the hailstone.
Thus, for example, two objects may be group isomorphic without being ring isomorphic, since the latter isomorphism selects the additional structure of the multiplicative operator.
Thus the standard disciplinary structure of philosophy was established as
" Thus these two distinctions can be applied not only to class structure within society but denomination and racial segregation within religion.
Thus rocky coastlines, especially in shallow water, are typically inimical to the survival of artifacts, which can be dispersed, smashed or ground by the effect of currents and surf, possibly ( but not always ) leaving an artifact pattern but little if any wreck structure.
Thus the welfare analysis of oligopolies is sensitive to the parameter values used to define the market's structure.
Thus, their energy levels stack up, creating band structure out of each atomic energy level.
Thus during his lifetime, Shoghi Effendi prepared for the election of the Universal House of Justice, by establishing a strong administrative structure at the local and national levels.
Thus, one may perform algebraic operations, because of the group structure, and one may talk about continuous functions, because of the topology.
Thus it is seen that a polyp is an animal of very simple structure, a living fossil that has not changed significantly for about half a billion years ( per generally accepted dating of Cambrian sedimentary rock ).
Thus Offa's Dyke is a combined structure and Car Dyke is a trench though it once had raised banks as well.
Thus two Elders and two Eldresses formed the Ministry at the top of the administrative structure.
Thus grammars that employ phrase structure rules are constituency grammars (= phrase structure grammars ), as opposed to dependency grammars, which view sentence structure as dependency-based.
Thus the spelling of a word is dependent principally upon its structure, its relationship to other words, and its language or origin.
He claimed that on top of individual exploitation, the capitalist also unjustly appropriates the additional value ( termed “ collective force ”) produced by co-operative activity as a “ force of one thousand men working twenty days has been paid the same wages that one would be paid for working fifty-five years ; but this force of one thousand has done in twenty days what a single man could not have accomplished, though he had laboured for a million centuries .” Thus the capitalist may have “ paid all the individual forces ” but “ the collective force still remains to be paid ” and which the capitalists “ enjoy unjustly .” Other notable anti-capitalist thinkers claim that the elite maintain wage slavery and a divided working class through their influence over the media and entertainment industry, educational institutions, unjust laws, nationalist and corporate propaganda, pressures and incentives to internalize values serviceable to the power structure, state violence, fear of unemployment and a historical legacy of exploitation and profit accumulation / transfer under prior systems, which shaped the development of economic theory:
Thus, in a sense, the notion of a Noetherian ring unifies the ideal structure of various " natural rings ".
Thus, a hexagonal structure uses the least material to create a lattice of cells within a given volume.

Thus and can
Thus, it is no mystical intuition, but an analyzable conception to say that man and his tradition can `` fall out of existence ''.
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, for aqueous media, we can think of the idealized organic active as an oleophilic or hydrophobic surface-active agent, and of an idealized builder as a oleophobic or hydrophilic surface-active agent.
Thus, the larvae of the intruder can develop at the expense of the rightful inhabitants and the store of beebread.
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, one can think of a dictionary entry as a word rather than a form.
Thus the unstressed it of it rarely snows here gets its significance from its use with snows: nothing can snow snow but `` it ''.
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, besides the training provided to youth in school, the existence of the school program can have supplementary benefits to industry which make it an asset to industrial development efforts.
Thus, for non-negative changes in the basic wage rate, the industry becomes the active wage-setter, since any increase in the basic wage rate can occur only by reason of industry acquiescence.
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, man can readily deduce that the primary objective end of the conjugal act is procreation, the propagation of the race.
Thus, while it remains possible that the Babylonians and/or the Pythagoreans may perhaps have had the magic square of three before the Chinese did, more definite evidence will have to turn up from the Middle East or the Classical World before China can lose her claim to the earliest known magic square by more than a thousand years.
" Thus Boolos and Jeffrey are saying that an algorithm implies instructions for a process that " creates " output integers from an arbitrary " input " integer or integers that, in theory, can be chosen from 0 to infinity.
Thus an algorithm can be an algebraic equation such as y
Thus, an algorithm can be considered to be any sequence of operations that can be simulated by a Turing-complete system.
Thus although most species in the order are herbaceous, some no more than 15 cm high, there are a number of climbers ( e. g. some species of Asparagus ), as well as several genera forming trees ( e. g. Agave, Cordyline, Yucca, Dracaena ), some of which can exceed 10 m in height.
For every group G there is a natural group homomorphism G → Aut ( G ) whose image is the group Inn ( G ) of inner automorphisms and whose kernel is the center of G. Thus, if G has trivial center it can be embedded into its own automorphism group.
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.
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:

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