Thus in the presence of large quantum computers an n-bit key can provide at least n / 2 bits of security.

Thus it can be interpreted without any reference to the zero-point energy ( vacuum energy ) or virtual particles of quantum fields.

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

" Thus irreal works such as Italo Calvino's Cosmicomics and Jorge Luis Borges ' Ficciones can be seen as an attempt to find a new allegorical language to explain our changed perceptions of the world that have been brought about by our scientific and technical culture, especially concepts such as quantum physics or the theory of relativity.

Thus if we wish to claim that no-cloning is a uniquely quantum result, some care is necessary in stating the theorem.

Thus the entire field of quantum physics emerged, leading to its wider acceptance at the Fifth Solvay Conference in 1927.

Thus, measuring a quantum state described by complex coefficients ( a, b ,..., h ) gives the classical probability distribution and we say that the quantum state " collapses " to a classical state as a result of making the measurement.

Thus, at least in the low-energy regime, the model is indeed a predictive quantum field theory.

Thus, the uncertainty principle actually states a fundamental property of quantum systems, and is not a statement about the observational success of current technology.

Thus the conquest of actual infinity may be considered an expansion of our scientific horizon no less revolutionary than the Copernican system or than the theory of relativity, or even of quantum and nuclear physics.

Thus, quantum physics casts reasonable doubt on the traditional determinism of classical, Newtonian physics in so far as reality does not seem to be absolutely determined.

Thus, the ontology of pilot wave theory contains as the trajectory we know from classical mechanics, as the wave function of quantum theory.

Thus a quantum mechanical ensemble is specified by a mixed state in general.

Thus, the quantum number has the value

Thus decoherence is the process by which information of a quantum system is altered by the system's interaction with its environment ( which form a closed system ), hence creating an entanglement between the system and heat bath ( environment ).

Thus, by expressing quantum mechanics in phase space ( the same ambit as for classical mechanics ), the Weyl map facilitates recognition of quantum mechanics as a deformation ( generalization ) of classical mechanics, with deformation parameter ħ / S, where S is the action of the relevant process.

Light's wavenumber is proportional to frequency, and therefore also proportional to light's quantum energy E. Thus,.

Thus, if the oscillator is in the quantum state it is also with nonzero probability in the other quantum state

Thus, the instanton field solution of the ( 1 + 1 )- dimensional field theory ( first quantized quantum mechanical system ) allows to be interpreted as a tunneling effect between the two vacua of the physical Minkowskian system.

Thus, the many-worlds quantum model is proved to be both real, and accessible, a fantastic achievement.

Thus, according to the theory, even the vacuum has a vastly complex structure and all calculations of quantum field theory must be made in relation to this model of the vacuum.

Thus, negative temperature is a strictly quantum phenomenon.

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, the alignment of the `` dots '' and `` tips '', respectively, indicate individual variability of the 21 growth centers of each child with respect to the mean values for these boys and girls.

Thus the Morse code " dots " and " dashes " are audible as beeping sounds.

Thus before the last ranks have fired, the first will have reloaded, as the following diagram shows: these little dots:: show the route of the ranks as they leave after firing.

Thus the dots below numbers do not always vertically align with each other, since some of them may be moved slightly downward so as not to collide with the note length lines.

Thus when a child draws a picture of a cat, they will always draw the same basic image, perhaps modified ( this cat has stripes that one has dots, for example ).

Thus at the same time that William Henry Harrison was preparing to pacify the aborigines of Indiana Territory and winning fame at the battle of Tippecanoe, Anglo-Saxon settlement made a great leap into the center of the North American continent to the west of the American agricultural frontier.

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, from the human body, the usual form assigned to the Deity, forasmuch as it is written that God created man in his own image, issue the two supporters, Nous and Logos, symbols of the inner sense and the quickening understanding, as typified by the serpents, for the same reason that had induced the old Greeks to assign this reptile for an attribute to Pallas.

Thus numbers like 2 and 120 ( 2 × 60 ), 3 and 180 ( 3 × 60 ), 4 and 240 ( 4 × 60 ), looked the same because the larger numbers lacked a final sexagesimal placeholder.

Thus, services could be moved to a different machine and, so long as they kept the same service name, there was no need for users to do anything different to continue accessing the service.

Thus, many users of a given BBS usually lived in the same area, and activities such as BBS Meets or Get Togethers were common, where users of the board would gather at a local restaurant, the SysOp ’ s home or similar venue and meet face to face.

Thus, any error in technology or judgment may have far-reaching consequences ( though the same can also happen through natural reproduction ).

Thus, if two species use the same codon at the same place to specify an amino acid that can be represented by more than one codon, that is evidence for a recent common ancestor.

Thus, a 3072-bit Diffie-Hellman key has about the same strength as a 3072-bit RSA key.

Thus, Abraham, Moses, David, and all Christians today are saved by the same faith.

Thus, covalent bonding does not necessarily require the two atoms be of the same elements, only that they be of comparable electronegativity.

Thus, for example, the same equation can be made to produce both a lyrical and melodic piece of music in the style of the mid-nineteenth century, and a fantastically dissonant cacophony more reminiscent of the avant-garde music of the 1950s and 1960s.

Thus, for example, the CGS unit of pressure, barye, is related to the CGS base units of length, mass, and time in the same way as the SI unit of pressure, pascal, is related to the SI base units of length, mass, and time:

Thus one can be tried twice for the same alleged crime.

Thus the probability of two failures in a same RAID group in time proximity is much smaller ( approximately the probability squared, i. e., multiplied by itself ).

Thus it works as a separation sign and not as an indication for an alternative version of the i. Diacritics can be used for emphasis ( érg koud for very cold ) or for disambiguation between a number of words that are spelled the same when context doesn't indicate the correct meaning ( één appel

Thus the probability of two failures in a same RAID group in time proximity is much smaller ( approximately the probability squared, i. e., multiplied by itself ).

Thus, interlaced video captures samples the scene motion twice as often as progressive video does, for the same number of frames per second.

Thus the dragoon regiments of the Imperial German Army wore the pickelhaube ( spiked helmet ) of the same design as those of the infantry and the British dragoons wore scarlet tunics, In other respects however dragoons had adopted the same tactics, roles and equipment as other branches of the cavalry and the distinction had become simply one of traditional titles.

Thus, several civilizations may come and go during the lifespan of one and the same planet.

Thus, if n < sub > r </ sub > is the average number of times a new civilization reappears on the same planet where a previous civilization once has appeared and ended, then the total number of civilizations on such a planet would be ( 1 + n < sub > r </ sub >), which is the actual reappearance factor added to the equation.

Thus, a 7A is a common jazz stick with a wooden tip, while a 7N is the same weight of stick with a nylon tip, and a 7B is a wooden tip but with a different tip profile, shorter and rounder than a 7A.

Thus an equivalence relation over X, a partition of X, and a projection whose domain is X, are three equivalent ways of specifying the same thing.