Thus, at pH between 2. 2 and 9. 4, the predominant form adopted by α-amino acids contains a negative carboxylate and a positive α-ammonium group, as shown in structure ( 2 ) on the right, so has net zero charge.

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 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, 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 initially the Level 2 cache ran at half of the CPU clock speed up to 700 MHz ( 350 MHz cache ).

Thus, for example, fumaroles from the Laguna Caliente crater of Poás Volcano create extremely high amounts of acid rain and fog, with acidity as high as a pH of 2, clearing an area of any vegetation and frequently causing irritation to the eyes and lungs of inhabitants in nearby settlements.

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, after a bid of 3, bids of 2 or 3 are not sufficient, but 3 or 4 are.

Thus, if the score was 3 – 2 with the bases loaded in the bottom of the ninth, and the batter smacked an " over the fence home run ", the game would end at 4 – 3, with the batter only allowed a double, and the runners officially stopped on 2nd and 3rd ( since they were not needed to win the game ).

Thus, an embosser can cost roughly anywhere from US $ 2, 000 to $ 80, 000, depending on the user's needs.

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

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 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 the GCD of 48 and 180 is 2 × 2 × 3 = 12.

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 the wavelength of a 100 MHz electromagnetic ( radio ) wave is about: 3 × 10 < sup > 8 </ sup > m / s divided by 10 < sup > 8 </ sup > Hz = 3 metres.

Thus for example the calculation 34 × 13 could be computed using the grid

Thus, for example, if the length of the equatorial segment on the diagram is 0. 4 times the length of the analemma on the diagram, then the segment in the celestial analemma would subtend 0. 4 × 47 or 18. 8 degrees at the observer on the ground.

Thus, negating one column if necessary, and noting that a 2 × 2 reflection diagonalizes to a + 1 and − 1, any orthogonal matrix can be brought to the form

Thus, after binding energy has been removed, binding energy = mass change × c < sup > 2 </ sup >.

Thus the Astrocade system was left with just the lower resolution 160 × 102 mode.

Thus a 2 × 2 matrix is symplectic iff it has unit determinant.

Thus, each X < sub > i </ sub > is a k < sub > i </ sub >- columned submatrix of X. Matrix Β has size k × m, and each of its columns consists of the components of vectors β < sub > i </ sub > and zeros, depending on which of the regressors from X were included or excluded from X < sub > i </ sub >.

Thus, for the whole Earth ( which has a cross section of 127, 400, 000 km² ), the power is 1. 740 × 10 < sup > 17 </ sup > W, plus or minus 3. 5 %.

Thus the Sun emits about 2. 2 billion times the amount of radiation that is caught by Earth, in other words about 3. 86 × 10 < sup > 26 </ sup > watts.

Thus a long ton is 20 × 8 × 14 lb

Thus P is a smooth manifold with a smooth map π: P → M which looks locally like the trivial bundle M × H → M. The frame bundle of M is a principal GL ( n )- bundle, while if M is a Riemannian manifold, then the orthonormal frame bundle is a principal O ( n )- bundle.

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 for instance the determinant of a matrix with integer coefficients will be an integer, and the matrix has an inverse with integer coefficients if and only if this determinant is 1 or − 1 ( these being the only invertible elements of the integers ).

Thus, for a substitution tiling the eigenvalues of the substitution matrix should be Pisot numbers.

Thus, matrix vesicles convey to the extracellular matrix calcium, phosphate, lipids and the annexins which act to nucleate mineral formation.

Thus the act of measurement induces a fundamental irreversible change on the density matrix ; this is analogous to the " collapse " of the state vector, or wavefunction collapse.

Thus, W is the negative of its transpose, which implies it is a skew symmetric matrix.

Thus before you did in the first example modifies the matrix clause Fred arrived.

Thus the elements of second row of matrix are identical to those of the first row and hence the third and fourth rows etc.

Thus the elements of second row of matrix are not identical to that of the first row, neither are the third and fourth rows, etc.

Thus the system can be described by a collection of coefficients known as the elastance matrix or reciprocal capacitance matrix, which is defined as:

Thus the matrices must be ordered appropriately, with the last matrix premultiplying the second last, and so on until the first matrix is premultiplied by the second.

Thus every rotation can be represented uniquely by an orthogonal matrix with unit determinant.

Thus a unitary structure can be seen as an orthogonal structure, a complex structure, and a symplectic structure, which are required to be compatible ( meaning that one uses the same J in the complex structure and the symplectic form, and that this J is orthogonal ; writing all the groups as matrix groups fixes a J ( which is orthogonal ) and ensures compatibility ).

Thus, the matrix D = ( d < sub > i, j </ sub >) with

Thus, in mathematical terms, they never appear as indices to the scattering matrix, which is to say, they never appear as the observable inputs and outputs of the physical process being modelled.

Thus the change of basis matrix in going from the original basis to the reciprocal basis is

Thus, each row ( or column ) of a stochastic matrix is a probability vector, which are sometimes called stochastic vectors.

Thus, if the information of the sub-matrices is available, the eigenvalues of the matrix can be described using that kind of information.

Thus, each different type of atom can be identified in the matrix as the tip is moved across the surface.