Summation, 540 symbols used

Note that the integrals replace the summation symbols used in the definition of M (Eq. 5.3). Also,... 

Fig. 4.1 Block diagram of a closed-loop control system. R s) = Laplace transform of reference input r(t) C(s) = Laplace transform of controlled output c(t) B s) = Primary feedback signal, of value H(s)C(s) E s) = Actuating or error signal, of value R s) - B s), G s) = Product of all transfer functions along the forward path H s) = Product of all transfer functions along the feedback path G s)H s) = Open-loop transfer function = summing point symbol, used to denote algebraic summation = Signal take-off point Direction of information flow.

We can represent a sum of terms using a shorthand notation involving the summation symbol Z. For example, the sum of terms... 

With respect to tiie symbol E(k=0,N), let us use from now on a general expression like E(k=r,s) as a sequence of sums and borrowing a programming terminology let us call it a nested summation symbol. 

Nested summation symbolism is very convenient for practical purposes, because successive generation of k vector elements can be programmed in a general way imder any high level language, using a unique do or for loop irrespective of the number of sums, or binomial expressions of type (3.4) entering the product (3.5). 

Nested Summation Symbols are ideal constructs for programming in a suitable hardware environment, for example using transputer boards [51] or any computing device with various CPU runnixig in parallel [62]. All operations attached to each vector k can be run in a separate CPU. Counting the times this can be done in... 

For the two terms in the sum-over-states expression in Eq. (42) that involve the ground state = 0, the transition frequency a) o is zero. The two terms are of opposite sign and will therefore cancel, and it is common practice to exclude the ground state from the summation and to use a primed summation symbol for the sum over excited states. 

Equation (16.25) can be simplified further if we use the summation convention, that is, if we omit the summation symbol from the equation, assuming that repeated symbols are summed from 1 to 3. As a result, the continuity equation becomes... 

This integral is equivalent to the area under the curve in Fig. 8 between 400 and 700 nm. The integral suggest that /ph,a can be accurately described by a mathematical equation. In reality this is not always evident and this integral therefore is commonly solved numerically based on the measured Fph,a over small wavelength intervals (AA), preferably as small as 1 nm. Mathematically this can be described by using the summation symbol S according to ... 

The summation symbol suggests that is a discrete set of functions. This need not be true. Contributions from functions whose eigenvalues are in a continuum would require integration rather than summation. However, we will use the sum symbol since most actual applications of perturbation theory in quantum chemistry invoke only discrete functions. 

The symbol M represents the masses of the nuclei in the molecule, which for simplicity are taken to be equal. The symbol is the Kionecker delta. The tensor notation is used in this section and the summation convention is assumed for all repeated indexes not placed in parentheses. In Eq. (91) the NACT appears (this being a matrix in the electronic Hilbert space, whose components are denoted by labels k, m, and a vector with respect to the b component of the nuclear coordinate R). It is given by an integral over the electron coordinates... 

Fig. 2.2. Average electrostatic potential mc at the position of the methane-like Lennard-Jones particle Me as a function of its charge q. mc contains corrections for the finite system size. Results are shown from Monte Carlo simulations using Ewald summation with N = 256 (plus) and N = 128 (cross) as well as GRF calculations with N = 256 water molecules (square). Statistical errors are smaller than the size of the symbols. Also included are linear tits to the data with q < 0 and q > 0 (solid lines). The fit to the tanh-weighted model of two Gaussian distributions is shown with a dashed line. Reproduced with permission of the American Chemical Society...

The spherical terms of the charge distribution contribute only to the diagonal elements of the even moments. Evaluation of these contributions is further discussed in section 7.2.3. For the higher-order terms in the summation, we have, using the symbol Oj for the jth moment operator ... 

The symbol will thus be used to indicate a soft coordinate and 2 to indicate a coordinate that could be either soft or hard. The ranges of summation for each type of index will be those indicated above, and summation over repeated indices will be implicit, unless stated otherwise. 

Here the <5 s are Kronecker symbols and the primed summations extend only over nonvanishing k-vectors. In deriving this result we used the orthonormality of the plane waves Q-l/2e-ifcr... 

J. Cizek, F. Vinette, and E. J. Weniger, Int. ]. Quantum Chem., Quantum Chem. Symp., S25, 209 (1991). Examples on the Use of Symbolic Computation in Physics and Chemistry Applications of the Inner Projection Technique and of a New Summation Method for Divergent Series. 

The summation over Jr, is then completed using the orthogonality of the Wigner 6j symbols,... 

Tensors are denoted by bold-face symbols, 0 is the tensor product, and the scalar product. For example, with respect to a Cartesian basis e AB = AikBkjei ej,A B = AijBij, and C B = C hlBkiei 0 ej, with summation implied over repeated Latin indices. The summation convention is not used for repeated Greek indices. 

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... 

This equation may seem quite complicated at first, but it is easier to apply once we understand the meaning of all the symbols. First of all, x, stands for the individual measurements made. There are 5 values of x, so i takes the values 1, 2, 3, 4, and 5. We thus have, in the first set of data, xi = 21, X2 = 24, X3 = 25, X4 = 26, and xs = 29. The quantity x is the average of these, which we have already established as 25. The number of data is , in this case 5. The symbol S (capital Greek sigma) represents a summation, so we simply add all the terms which appear to the right of it. It is easiest to do this using a table, as shown below. 

The use of a bar over a symbol indicates that the average is meant—thus X is the average of the set of x s considered. S indicates summation, i.e. addition, of the series of terms considered. 

We have used that [ "] = E and the fact that Wp M2, and 0)3 are dummy summation indices that run over both positive and negative frequencies. None of Eqs. (59)-(62) is symmetric in the tensor indices y, and 8. As pointed out in connection with Eq. (7), we normally choose our hyperpolarizability tensors to possess intrinsic symmetry, and it is clear that we can accomplish this without changing the polarization of the molecule by taking the average of the six terms generated by permuting pairs of the dummy indices (j3, Wj), (% M2), and (8, 0)3) we denote this operation with the symbol l/6J2 i,2,3> where tlie factor of one sixth is required to maintain the same value of the polarization. The third-order polarization in Eq. (36) can then be written as... 

These coefficients, and the summations in Eq. (6.60), are most easily computed using a symbolic math program. Some examples of co, v, g) for selected moments up to fifth order are given in Tables 6.1-6.9. For clarity, in these tables we have denoted the velocity difference vector by g = gi,g2,g3) and g =g + gl+gy The final expressions for the collision source terms for integer moments of order j = h+h + h can now be written in the form of Eq. (6.54) as... 

This function is similar to the electron density function given earlier. Here, P(uvw) is the value of the Patterson function at Patterson coordinates u, v, w these are the traditional coordinate symbols (instead of x, y, z) used for squared ( F jt,p) space. All other symbols have their usual meaning. The Patterson function is a Fourier summation using the intensities as coefficients and setting all equal to 0. The resulting contoured map will have peaks corresponding to vector differences between all atoms in the structure. A vector between an atom and itself is a zero vector therefore, the Patterson functions always have a very large peak at u,v,w = 0, 0, 0. 

Note that this is in terms of angles. Firstly, to avoid confusing the symbol x with distance (distance being also often called x ), it helps to first change it to a more familiar symbol for angle, like 0. We can then also use the summation sign to write the above expansion in a more compact form ... 

The symbol Jy is often used to represent this Coulomb interaction between electrons in spin orbitals i andand is unfavourable (i.e. positive). The total electrostatic interaction between the electron in orbital Xi and the other N — 1 electrons is a sum of all such integrals, where the summation index j runs from 1 to N, excluding i ... 

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