Sum of differences

It does not help to discuss the influence of substituents in nitro substrate (187) on the stereoselectivity and stereodirection of [3+ 2]-cycloaddition in the general case, because this effect is determined by the sum of different factors and, hence, it depends on the number and nature of the above mentioned substituents. 

There are several reasons why the sum of squares, i.e. the sum of squared differences between the measured and modelled data, is used to define the quality of a fit and thus is minimised as a function of the parameters. It is instructive to consider alternatives to the sum of squares, (a) Minimal sum of differences - is not an option, as positive and negative differences cancel each other out. Huge deviations in both directions can result in zero sums. 

The Hamiltonian operator in Eq. 1 contains sums of different types of quantum mechanical operators. One type of operator in Ti gives the kinetic energy of each electron in by computing the second derivative of the electron s wave function with respect to all three Cartesian coordinates axes. There are also terms in H that use Coulomb s law to compute the potential energy due to (a) the attraction between each nucleus and each electron, (b) the repulsion between each parr of electrons, and (c) the repulsion between each pair of nuclei. 

Figure 16.19—ICP - MS recording plasma torch ionisation for elemental analysis. A spectrum of a sample containing metals in the mass range of 42 to 68 Da is shown. The sample contains a high concentration of calcium that constitutes a major peak at 40 Da (not shown on the figure). The resolution in this recording is not sufficient to separate isobaric ions, thus peaks in the spectrum are the sum of different species with the same nominal mass.

AOT40 18,000 pg/ m3 h For environmental protection Sum of differences between 8 am and 8 pm of hourly value exceeding 80 pg/m3 between May and July over 3 years... 

Analytes on the macroscopic scale are compounds, elements, species. On the microscopic scale the identity is given by the electronic, nuclear or molecular structure of the particles [2], On the macroscopic scale the analyte is in the most cases a sum of different identities. 

Actually HMS is a sum of different interaction operators each related to an interaction with a different physical origin. The coupling between interactions is ensured by the iterative solution of the pseudo HF (or DFT) solution of the whole Schrodinger equation. These operators are expressed in terms of solvent response functions based on an averaged continuous solvent distribution. They will be symbolically indicated with the symbols Qx(r, r ) where r is a position vector and x stands for one of the interactions. We shall examine later the form of some of the operators, which actually are the kernels of integral equations. 

Besides, examination of Eq. (201) shows that the spectral density is the sum of different components, each of them being a superposition of Lorentzians involving different half-widthes and intensities. Note that this result differs deeply from the situation without indirect damping given by Eq. (82), for which all the Lorentzians forming the line shape have the same half-width. This Ending may be verified by taking y = 0 in Eq. (201). 

The final uncertainty of the fit was calculated as a random sum of differences of the fits without function h and with h Z) from the binomial expansion of the expression... 

In order to derive the equation for pure reaction overpotential we assume that j0 is high, i.e., an equilibrium exists at the electrode metal surface in respect of the surface concentrations of O and R. All other steps including diffusion are fast enough, i.e., much faster than reaction (2). (In general, r is a sum of different contributions [- activation overpotential, - diffusion overpoten-tial etc.] which might be interdependent.)... 

Personal preference often dictates which of these measures of dispersion is utilized. Relative standard deviation is probably the most widely used. Variance finds its greatest application in calculations of the most probable uncertainty in the case of sums of differences, which is found by taking the square root of the sum of the individual variances. It is an advantage to use a over cr2 for expressing precision because the units of a are the same as that of the quantity measured. 

If the relaxation curve is a sum of different exponentials (corresponding to different sites), there are N different constants C, but when plotted as normalized vs (jiT), curves taken at different temperatures collapse into one (84), as shown in Fig. 45. It is usually impossible to determine a value of N from the data. Most relaxation curves can be described by a sum of... 

In this approach, the time dependence is considered in the relaxation flux F,f. Grassi and coworkers [50] improved the Camera-Roda and Sard model assuming that F,r is the sum of different contributes, each one characterized by its proper relaxation time. In this manner, the description of membrane viscoelastic properties is more adherent to reality. 

To apply this method to mono-THF acetogenins, a "difference minimization" scheme similar to that earlier described was used. That is, the differences in chemical shifts between a model and the natural product for all relevant hydrogens were added, and this value was compared for each of the models. The model with the smallest sum of differences is concluded to have the same relative configuration as the natural product. 

Simple sums of different molecular descriptors were proposed as - superindices in order to obtain highly discriminant indices particular superindices were suggested to account for -> molecular complexity. 

Studies of the influences of procedural variables (sample mass, partial pressure of COj, heating rate, etc.) on the kinetics of calcite powder dissociation continue to be published [49-51]. Such kinetic observations are the sum of different reaction rates... 

Type II represents that in the global burden of ischaemia there are crises with and without anginal pain. This is why the concept of total ischaemic burden involving the sum of different periods of time with symptomatic (ST-segment changes plus pain) and asymptomatic (exclusively ST-segment changes) ischaemia was coined. 

The symmetry of an isolated atom is that of the full rotation group R+ (3), whose irreducible representations (IRs) are D where j is an integer or half an odd integer. An application of the fundamental matrix element theorem [22] tells that the matrix element (5.1) is non-zero only if the IR DW of Wi is included in the direct product x of the IRs of ra and < f. The components of the electric dipole transform like the components of a polar vector, under the IR l)(V) of R+(3). Thus, when the initial and final atomic states are characterized by angular momenta Ji and J2, respectively, the electric dipole matrix element (5.1) is non-zero only if D(Jl) is contained in Dx D(j 2 ) = D(J2+1) + T)(J2) + )(J2-i) for j2 > 1 This condition is met for = J2 + 1, J2, or J2 — 1. However, it can be seen that a transition between two states with the same value of J is allowed only for J 0 as DW x D= D( D(°) is the unit IR of R+(3)). For a hydrogen-like centre, when an atomic state is defined by an orbital quantum number , this can be reduced to the Laporte selection rule A = 1. This is of course formal, as it will be shown that an impurity state is the weighted sum of different atomic-like states with different values of but with the same parity P = ( —1) These states are represented by an atomic spectroscopy notation, with lower case letters for the values of (0, 1, 2, 3, 4, 5, etc. correspond to s, p, d, f, g, h, etc.). The impurity states with P = 1 and -1 are called even- and odd-parity states, respectively. For the one-valley EM donor states, this quasi-atomic selection rule determines that the parity-allowed transitions from Is states are towards np (n > 2), n/ (n > 4), nh (n > 6), or nj (n > 8) states. For the acceptor states in cubic semiconductors, the even- and odd-parity states labelled by the double IRs T of Oh or Td are indexed by + or respectively, and the parity-allowed transition take place between Ti+ and... 

Figure 19.1 Measurement set-up of an ion selective electrode (ISE). The potential of the membrane permeable to the ion varies with the concentration of the specific ion in the solution sample. The measured signal is the sum of different potentials generated at aU interfaces. Measurements are typically conducted by an ionometer. They are made at zero current. Under the equilibrium conditions the transfer of ions from the membrane into solution is equal to the transfer from the solution to the membrane. Manufacturers also supply more robust electrodes called combined electrodes, that include both external and ion selective electrodes in the same device. pH electrodes are of this type. Unfortunately the schematics are less clear due to the compact arrangement.

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