For the case of an object that is small compared with the radial distance to its axis of rotation, such as a tin can swinging from a long string or a planet orbiting in a circle around the Sun, the angular momentum can be expressed as its linear momentum,, crossed by its position from the origin, r. Thus, the angular momentum L of a particle with respect to some point of origin is

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, most knowledge of Roman Britain has derived from archaeological investigations, and the epigraphic evidence lauding the Britannic achievements of an Emperor of Rome, such as Hadrian ( r. 117 – 38 ) and Antoninus Pius ( r. 138 – 61 ), whose walls demarcated the northern borders of Roman Britain.

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, tangential speed will be directly proportional to r when all parts of a system simultaneously have the same ω, as for a wheel, disk, or rigid wand.

: and this exceeds unity by only about r /( 2N-2b ) if N-b is large compared to r. Thus the weight of evidence provided by the sight of a white shoe is positive, but is small if the number of ravens is known to be small compared to the number of non-black objects.

Thus the wave form transmitted from an antenna has no distortion in time: the only distorting factor is 1 / r.

Thus, if the user is initially at a distance D from the target, the remaining distance after the first submovement is rD, and the remaining distance after the nth submovement is r < sup > n </ sup > D.

The preceding two paragraphs are an expression in English which unambiguously defines a real number r. Thus r must be one of the numbers r < sub > n </ sub >.

Thus, Richard says, the real number r will not be included as any r < sub > n </ sub >, because the definition of r does not meet the criteria for being included in the sequence of definitions used to construct the sequence r < sub > n </ sub >.

Thus y is the largest integer less than the nth root of x, and r is the remainder.

Thus, a parent and child pair has a value of r = 0. 5, siblings have a value of r = 0. 5, a parent's sibling has r = 0. 25, and first cousins have r = 0. 125.

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 the set of strings of length k in the domain is exactly the set of such strings already enumerated.

Thus, if ( M, d ) and ( N, d ) are two metric spaces, and, then there is a constant k such that

Thus, the initial step corresponds to k =

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, the " kilo -" symbol, k, can be used to produce km, kg, and kW, ( kilometre, kilogram, and kilowatt ).

While this equation suggests that the activation energy is dependent on temperature, in regimes in which the Arrhenius equation is valid this is cancelled by the temperature dependence of k. Thus, E < sub > a </ sub > can be evaluated from the reaction rate coefficient at any temperature ( within the validity of the Arrhenius equation ).

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

Thus, the DCT-I corresponds to the boundary conditions: x < sub > n </ sub > is even around n = 0 and even around n = N-1 ; similarly for X < sub > k </ 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, E ( k ) dk represents the contribution to the kinetic energy from all the Fourier modes with k < | k | < k + dk, and therefore,

Thus, every element of Dic < sub > n </ sub > can be uniquely written as a < sup > k </ sup > x < sup > j </ sup >, where 0 ≤ k < 2n and j = 0 or 1.

Thus, if k and n are coprime, z < sup > k </ sup > is also a primitive nth root of unity, and therefore there are φ ( n ) ( where φ is Euler's totient function ) distinct primitive nth roots of unity.

Thus, given the field K, any algebra can be specified up to isomorphism by giving its dimension ( say n ), and specifying n < sup > 3 </ sup > structure coefficients c < sub > i, j, k </ sub >, which are scalars.

Thus, in mathematical physics, the structure coefficients are often written c < sub > i, j </ sub >< sup > k </ sup >, and their defining rule is written using the Einstein notation as

Thus, the DST-I corresponds to the boundary conditions: x < sub > n </ sub > is odd around n =- 1 and odd around n = N ; similarly for X < sub > k </ sub >.