Subscripts mathematical expressions

The subscript x, y, or z on a 2p orbital indicates that the angular part of the orbital has its maximum value along that axis. Graphs of the square of the angular part of these three functions are presented in Figure 6.2. The mathematical expressions for the real 2p and 3p atomic orbitals are given in Table 6.2. 

We will now derive explicit mathematical expressions from which the curves in Figs. 11 and 12 are derived. We consider first a two mirror system where the amplitudes of reflection and transmission are given by r, and f, respectively, where the subscript i indicates mirror 1 or mirror 2. For mirrors of high reflectivity, there will be many reflections within the interferometer that will cause the apparent beam radius to grow. This effect is shown in Fig. 12b, which demonstrates how the beam radius grows with each round trip in the interferometer. We will account for this effect quantitatively in the sequel. 

Equation 2.17 is the mathematical expression of Student s conclusions. The new random variable is represented by the s anbol tn-i, and its distribution is called the t distribution or Student s distribution. The subscript n - 1 reminds us that the form of the distribution depends on the size of the sample. Actually there are several distributions, each one corresponding to a certain number of degrees of freedom involved in determining the value of s. 

The subscript notation 7f i) in the mathematical expression given in Eq. (1.40) is based on the mathematical theory of permutations [101], where the permutation function value n i) gives the rank of the (th molecule and the unique inverse r j) designates they th molecule in the overall ranking. A graphic example of how these functions operate is provided in Fig. 1.5. It is important to note that while the permutations determine the rank order of the compormds, it is the similarity values themselves that are combined using the MEAN fusion rule in similarity fusion. 

In general, when very few samples are available the force Fk(N) will not be an accurate approximation of d/l/d . Large variations in Fj (N) may lead to nonequilibrium effects and systematic bias of the calculation. Mathematically, this can be expressed by introducing a perturbation zLif fq, p, N), which is a function of the number of steps N. At N = 1 if we average over all possible initial configurations, abbreviated by subscript 0, we obtain... 

Here it has been possible to drop the subscripts and 1 since the formula is the same for both when = 0. Also since the formulae for t O have the same mathematical form as the formula for negligible fields, we can write down expressions for the frequency of maximum absorption when f 7 0. For this we note that... 

The symbols used to denote units are printed in roman font those denoting physical quantities or mathematical variables are printed in italics and should generally be single letters that may be further specified by subscripts and superscripts, if required. The unit of any physical quantity can be expressed as a product of the SI base units, the exponents of which are integer numbers, e.g. [ ] = m2 kg s 2. Dimensionless physical quantities, more properly called quantities of dimension one, are purely numerical physical quantities such as the refractive index n of a solvent. A physical quantity being the product of a number and a unit, the unit of a dimensionless quantity is also one, because the neutral element of multiplication is one, not zero. 

As the wavelength is much larger than the dimension of the molecule (e.g., 5000 A with respect to 10 A) the space-depending factors can be put equal to one, meaning that they are constant over the whole molecule. However, the mathematical treatment becomes more subtle when it comes to introducing the absorption coefficient k (that will later be related to the molar absorption coefficient e). A subscript has to be introduced, meaning that at the absorption frequency v y corresponding to the frequency of the transition from level a to level j or in a close vicinity, the expressions (e " / ) and (e / ) have a fixed value (see Piepho and Schatz, 1983, pp. 11 and 17). Within the frame of the fixed coordinate system of a molecule, the space-dependent factors may be expressed as a... 

The preceding argument shows that observers in dUfer-ent coordinate systems perceive different accelerations. Mathematically, this is expressed in the following way. First of all, the relationship between the derivative of the position vector r in an inertial reference ftame denoted by subscript 1 and its derivative in a reference frame rotating with angular velocity n relative to the inertial frame is... 

Ttie subscript descriptions of the d orbitals are chosen because such expressions are an important part of their respective mathematical formulations, representing the Cartesian component of the angular wave function, just as the 2p orbitals have the subscripts x, y and z. 

In the mathematics of logarithmic and exponential numbers, log means logarithm. The subscript 10 means base lO. Anytime we need to evaluate an expression of the form Iogj (i0 ), were a is a number, the answer is logJiO") = a. 

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