" 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, 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 < sub > 1 </ sub >-− 3 c < sub > 1 </ sub > will cause them to be renumbered from 0 through 9 and 4 through 23, respectively.

Thus, it has a year 0, the years before that are designated with negative numbers and the years after that are designated with positive numbers.

Thus ℵ < sub > 0 </ sub > is a regular cardinal.

Thus, r < sub > k </ sub > is smaller than its predecessor r < sub > k − 1 </ sub > for all k ≥ 0.

Thus an hour was 144 conventional minutes ( more than twice as long as a conventional hour ), a minute was 86. 4 conventional seconds ( 44 % longer than a conventional minute ), and a second was 0. 864 conventional seconds ( 13. 6 % shorter than a conventional second ).

#* Proof: Then so Thus However, is prime, so or In the former case, hence ( which is a contradiction, as neither 1 nor 0 is prime ) or In the latter case, or If however, which is not prime.

Thus the set of all polynomials with coefficients in the ring R forms itself a ring, the ring of polynomials over R, which is denoted by R. The map from R to R sending r to rX < sup > 0 </ sup > is an injective homomorphism of rings, by which R is viewed as a subring of R. If R is commutative, then R is an algebra over R.

e. g. The square-free number 42 has factorisation 2 × 3 × 7, or as an infinite product: 2 < sup > 1 </ sup > · 3 < sup > 1 </ sup > · 5 < sup > 0 </ sup > · 7 < sup > 1 </ sup > · 11 < sup > 0 </ sup > · 13 < sup > 0 </ sup > · ...; Thus the number 42 may be encoded as the binary sequence < tt >... 001011 </ tt > or 11 decimal.

Thus, p / q is the root of qx − p = 0.

Thus the bound vector represented by ( 1, 0, 0 ) is a vector of unit length pointing from the origin along the positive x-axis.

Thus, in terms of abundances, only about 0. 54 % of Vega consists of elements heavier than helium.

Thus the origin of the original space can be found at ( 0, 0, ... 0, 1 ).

Thus, the RF bandwidth of a signal ( measured from the lowest frequency as opposed to 0 Hz ) is twice its baseband bandwidth.

Thus, if a " 1 " is represented by one transition, then a " 0 " is represented by two transitions and vice versa, making Differential Manchester a form of frequency shift keying.

Thus, the maximum frequency deviation is </ br > < big > = 0. 25 f < sub > m </ sub >, </ big ></ br > where f < sub > m </ sub > is the maximum modulating frequency.

Thus, if a person owns 100 shares and the cash dividend is USD $ 0. 50 per share, the holder of the stock will be paid USD $ 50.

Thus, in the example, if Lloyds TSB are trading at 411p, then for every day I keep the bet open I am charged finance cost to be 7 % (( 411p x 10 ) * 7 % / 365 ) = £ 0. 78821 ( or 78. 8p )

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

Thus, to cite but one example, the Pax Britannica of the nineteenth century, whether with the British navy ruling the seas or with the City of London ruling world finance, was strictly national in motivation, however much other nations ( e.g., the United States ) may have incidentally benefited.

Thus we are compelled to face the urbanization of the South -- an urbanization which, despite its dramatic and overwhelming effects upon the Southern culture, has been utterly ignored by the bulk of Southern writers.

Thus Faulkner reminds us, and wisely, that the `` new '' South has gradually evolved out of the Old South, and consequently its agrarian roots persist.

Thus, the avant-garde choreographers have extended the scope of materials available for dance composition.

Thus, there is freshness not only in the individual movements of the dance but in the shape of their continuity as well.

Thus, paradoxically, the beat writers resort to `` religious '' metaphors: they are in search of mana, the spiritual, the numinous, but not anything connected with formal religion.

Thus jazz is transmuted into something holy, the sacred road to integration of being.

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

Thus the image of man has suffered complete fragmentation in personal and spiritual qualities, and complete objectification in sub-human and quasi-mechanistic powers.

Thus the fictional detective is much more than a simple businessman.

Thus the transformation of Adam Smith's ideal entrepreneur into a mythological detective coincides closely with the decline of the real entrepreneur in economic life.

Thus the cocktail party would appear to be the ideal system, but there is one weakness.

Thus, if corporations are not to run away with us, they must become quasi-governmental institutions, subject to public control and needs.

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 with regard to the loss of tradition, in the change from order to disorder the metaphysics of change works itself out as a disruption of the individual soul, a change in which man continues as an objective ontological existent, but no longer as a man.

Thus, circular motion is itself one of the essential characteristics of completely perfect celestial existence.

Thus earth has fallen to the center of the universe.

Thus, Margenau remarks: `` A large number of unrelated epicycles was needed to explain the observations, but otherwise the ( Ptolemaic ) system served well and with quantitative precision.

Thus, in no ordinary sense of ' simplicity ' is the Ptolemaic theory simpler than the Copernican.

Thus he complains, with considerable justice, that the Tory writers have resorted to libel instead of answering his arguments.

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 the student of literature may sometimes find it helpful to classify a poem or an essay as being in idea or in ideal content or subject matter typical or atypical of its period.

Thus ideas like `` grace '', `` salvation '', and `` providence '' cluster together in traditional Christianity.