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, for example, a 2x6 rectangle and a 3x4 rectangle are equal but not congruent, and the letter R is congruent to its mirror image.

Thus every module direct summand of R is generated by an idempotent.

Thus, Hillel called attention to seven commonly used in the interpretation of laws ( baraita at the beginning of Sifra ); R. Ishmael, thirteen ( baraita at the beginning of Sifra ; this collection is largely an amplification of that of Hillel ).

Thus Euclidean geometry corresponds to the choice of the group E ( 3 ) of distance-preserving transformations of the Euclidean space R < sup > 3 </ sup >, conformal geometry corresponds to enlarging the group to the conformal group, whereas in projective geometry one is interested in the properties invariant under the projective group.

Thus, on 10 February of that year, a Junker 52 of the Réseau des Liaisons Aériennes Francaises ( R. L. A. F ), later known as Air France, landed in Plaisance.

Thus, as from 1945, the R. L. A. F operated the Paris-Mauritius line.

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.

Thus, if R ≥ VC then firm should operate.

Thus the ( real < ref > The complex spinors are obtained as the representations of the tensor product H ⊗< sub > R </ sub > C = Mat < sub > 2 </ sub >( C ).

CSMA / CA can optionally be supplemented by the exchange of a Request to Send ( RTS ) packet sent by the sender S, and a Clear to Send ( CTS ) packet sent by the intended receiver R. Thus alerting all nodes within range of the sender, receiver or both, to not transmit for the duration of the main transmission.

Thus these are precisely the class of domains for which Frac ( R )/ Prin ( R ) forms a group, the ideal class group Cl ( R ) of R. This group is trivial if and only if R is a PID, so can be viewed as quantifying the obstruction to a general Dedekind domain being a PID.

Thus L ⊆ NL ⊆ P. The inclusion of L into P can also be proved more directly: a decider using O ( log n ) space cannot use more than 2 < sup > O ( log n )</ sup > = n < sup > O ( 1 )</ sup > time, because this is the total number of possible configurations.

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 < sub > 2 </ sub > 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 < sub > 1 </ sub >-− 3 c < sub > 1 </ sub > will cause them to be renumbered from 0 through 9 and 4 through 23, respectively.

Thus this class of molecules belong to the D < sub > 2d </ sub > point group.

Thus, few drops of diamminesilver ( I ) hydroxide ( Ag ( NH < sub > 3 </ sub >)< sub > 2 </ sub > 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 < sub > 3 </ sub >– CH < sub > 2 </ sub >– OH ), separating the functional group from another part of the molecule ( C < sub > 2 </ sub > H < sub > 5 </ sub > OH ), or by its atomic constituents ( C < sub > 2 </ sub > H < sub > 6 </ sub > O ), according to what is discussed.

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

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 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, 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, 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.

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, 0 dBrn =-90 dBm.

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.