In particular, one should think of the condition number as being ( very roughly ) the rate at which the solution, x, will change with respect to a change in b. Thus, if the condition number is large, even a small error in b may cause a large error in x.

Thus, any other number c that divides both a and b must also divide g. The greatest common divisor g of a and b is the unique ( positive ) common divisor of a and b that is divisible by any other common divisor c.

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 ' b ' and'd ' can rhyme ( both being ' voiced plosives '), as can ' bh ' and ' l ' ( which are both ' voiced continuants ') but ' l ', a ' voiced continuant ', cannot rhyme with ' ph ', a ' voiceless continuant '.

Thus, the units for b are time / bit ; e. g., ms / bit.

Thus, the difference of any two solutions to the equation Ax = b lies in the null space of A.

Thus the letters sh, sh, ch become b, b, l, i. e., Babel ( see: Atbash cipher ). This code is called the Atbash Cipher and was used to decrypt.

Thus, New Jersey has a law which makes it a “ disorderly persons offense ” to knowingly to engage in the unauthorized practice of law, and a “ crime in the fourth degree ” to commit UPL if one ( a ) creates a false impression that one is a lawyer ; ( b ) derives a benefit from UPL, or ( c ) causes an injury by UPL.

Thus, if P ( a, b, c ) is the predicate " a · b = c " and is the set of natural numbers, then

Thus he argued that the soul was a material substance, and that this was proved ( a ) by the circumstance that not only bodily qualities, but also mental capacity, are transmitted by ordinary generation from parent to child ; and ( b ) by the sympathy of the soul with the body seen in the fact that, when the body is struck or cut, the soul is pained ; and when the soul is torn by anxiety or depressed by care, the body is correspondingly affected.

Thus if U < sub > p </ sub >, V < sub > p </ sub >, Y < sub > p </ sub > are three tangent vectors at p and a and b are real numbers, then

Thus for a spherical galaxy with a equal to b, the number

Thus, the number R < sub > n </ sub >< sup >( b )</ sup > consists of n copies of the digit 1 in base b representation.

Thus, a pocket veto can only occur under two circumstances: a ) if Congress is adjourned, and b ) if bill return to Congress is not possible ( bill return is possible when Congress has adjourned, but has designated an agent to receive veto messages and other communications, an action Congresses have taken routinely for decades ).

: Thus, the MDCT of 2N inputs ( a, b, c, d ) is exactly equivalent to a DCT-IV of the N inputs: (– c < sub > R </ sub >– d, a – b < sub > R </ sub >), where R denotes reversal as above.

Thus, a and b are the eigenvalues corresponding to u and v, respectively.

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 if a represents Socrates then Phil ( a ) asserts the first proposition, p ; if a instead represents Plato then Phil ( a ) asserts the second proposition, q.

Thus there is a number x ≡ ( q − 1 )< sup >( p − 2 )</ sup > for which ( q − 1 )· x ≡ 1 ( mod p ), and therefore a number k for which ( q − 1 )· x − 1 = kp.

Thus 2 < sup >( q − 1 ) x </ sup >/ 2 < sup > kp </ sup >

Thus the initial assumption that p and q − 1 are relatively prime is untenable.

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

Thus, F (− q ) has the same magnitude as F ( q ) but they have the opposite phase, i. e., φ ( q ) = − φ ( q )

Thus for English the " e " box is the largest while the " j ", " k ", " q ", " x ", and " z " boxes are the smallest.

Thus the efficiency depends only on q < sub > C </ sub >/ q < sub > H </ sub >.

Thus, if one of the coordinates, say q < sub > k </ sub >, does not appear in the Lagrangian, the right-hand side of the equation is zero, and the left-hand side requires that

Thus, the absence of the ignorable coordinate q < sub > k </ sub > from the Lagrangian implies that the Lagrangian is unaffected by changes or transformations of q < sub > k </ sub >; the Lagrangian is invariant, and is said to exhibit a symmetry under such transformations.

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, q² ) depending on the author.

Thus denoting the common side as a and the diagonals as p and q, in every rhombus

Thus, according to Euler's quadrilateral theorem, the circumradius can be expressed in terms of the diagonals p and q, and the distance x between the midpoints of the diagonals as

Thus, this picture can be taken as a picture of eta as a function of q.

Thus the compound p → q is false if and only if both p is true and q is false.