Summation sign

Since under the summation sign each temr is zero (no coupling between the inside and the outside subspaces) [see Eq. (10)] we finally get... 

The forbidding appearance of the summation sign may be avoided merely by writing... 

The frequency associated with F n) is v . This frequency should be equal to n times the basis frequency, which is equal to l/(27 ) (this is the period of a sine or cosine which exactly fits in the measurement time). Thus v = n/(2T ) = n/(2NAt). It should be noted that in literature one may find other conventions for the normalization factor used in front of the integral and summation signs. 

To simplify the notation, it is customary to omit the summation sign and sum over indices which are repeated on the same side of the equation. An index which is not repeated is understood to take successively the values 1,2, so that there are altogether v different equations. With these... 

All the terms under the second summation sign with i f k vanish and therefore... 

Let us rewrite the term under the summation sign in a matrix form... 

For two reservoirs, the summation signs disappears from the descriptive equations (7.4.5). In addition, we can safely equate the kinetic and fractionation factors of 144Nd... 

The differentiations with respect t n h uml fi under the summation sign is justified by the fact that the series on the right-hand side of equation (13.2) is uniformly convergent for all real or complex values of h and ft which satisfy the relation [ ft 1, j h < -y/ 2 — 1. 

The summation sign has been included for emphasis. In the first line of (6.30) there do not appear to be any repeated subscripts, but that is because they have been subsumed in the abbreviated notation. If the stiffness tensor elements are written out in full, as in (6.24), this is immediately apparent. The selection rules in (6.30) correspond directly to the components of stress and strain that are related by each of the three stiffness constants. 

The sum is the sum of probabilities for a Poisson distribution with parameter Ax , which equals 1, so the term before the summation sign is the moment generating function, M(t) = cxp X(e -1)]. 

The prime on the summation sign indicates that the triple summation does not include the terms with both v = v and J = J. From (4.38), the integrals occurring in (4.54) are... 

The first factor under the summation sign takes care of the antisymmetry of the two-electron states of the shell. The submatrix elements are summed in accordance with the statistical weights of these states. Using (14.66), (6.25), (6.26) and (6.18), we sum the right side of (14.67) in the explicit form... 

If the angle (3 is much less than 1, then, in accord with Figs. 7 and 9, the most part of the rotators move freely under effect of a constant potential U0, since their trajectories do not intersect the conical cavity. A small part of the rotators moves along a trajectory of the type 1 shown in Fig. 10. However, at d > (3—that is, in the most part of such a trajectory—they are affected by the same constant potential U0- Therefore, for this second group of the particles the law of motion is also rather close to the law of free rotation. For the latter the dielectric response is described by Eq. (77). We shall represent this formula as a particular case of the general expression (51), in which the contributions to the spectral function due to longitudinal A) and transverse KL components are determined, respectively, by the first and second terms under summation sign. Free rotators present a medium isotropic in a local-order scale. Therefore, we set = K . Then the second term... 

In the special case that the limits of integration for the temperature and for the pressure are the same for each substance, the summation signs can be placed inside the integral signs. We can also define a quantity (ACP)P. as... 

A function of an operator is defined through its (Taylor) power series. The summation sign should really be understood as a summation over discrete quantum numbers and an integration over continuous labels corresponding to translational motion. 

Because dc may be brought outside the summation sign and is not zero, this gives as the general condition for chemical equilibrium ... 

When an equation is too long to fit on one line, break it after an operator that is not within an enclosing mark (parentheses, brackets, or braces) or break it between sets of enclosing marks. Do not break equations after integral, product, and summation signs after trigonometric and other functions set in roman type or before derivatives. 

In Eq. [20], Ri is the excess molar refraction (MR), which is the MR of the solute less the MR of the alkane with the same characteristic volume, V 2, as the solute. The 7t symbol is the dipolarity/polarizability, and X) P2 3re the so-called overall LIB acidity and basicity descriptors, respectively. The summation sign is used to emphasize that these are overall HB properties designed to be appropriate to situations where the solute molecule is surrounded by an excess of solvent molecules. These descriptors are in contrast to the HB descriptors 0.2 and p2 employed in Eq. [19], which are derived from 1 1 complexation constants. Equation [20] has also been used with the Vx2 term replaced by a log (L ) term, where is the equilibrium constant... 

The summation sign in the equation tells us to add up all the terms in the far right column, which gives 34. Note that the individual terms are all positive since the square of a negative number is positive. Substituting into the equation now gives... 

As noted in the previous equations, the summation sign S is used. Although only two points are needed to determine the constants, the actual experimentation may be conducted to produce several points. Since the equation only needs two points, the data are then grouped into two groups. Thus, / is the number of experimental data in the first group in a total of n experimental data points. The second gronp would consist of n - / data points. 

The notation 27 is well established in the literature. It is distinguished from a summation sign by its bold font and lack of summation limits. 

The prime symbol on the summation sign indicates that the term i —k or j = kis to be omitted from the summation. This first order differential equation (9.3.8) is ultimately solved for cpk by the standard technique of finding a multiplier k u) for both sides of (9.3.8) that converts the left into a differential function of the form... 

The summation bo is p) bo) and in the other summations the constants bo and bi can be removed from within the summation signs so that... 

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