Expectation values distinguishing

The electronic spatial extent is a single number that attempts to describe the size of a molecule. This number is computed as the expectation value of electron density times the distance from the center of mass of a molecule. Because the information is condensed down to a single number, it does not distinguish between long chains and more globular molecules. 

Anchored amine materials can be prepared through a number of synthetic methodologies. Because of the potential importance of these materials to organic synthesis, a ninhydrin assay was developed as a rapid laboratory determination of available surface amines. The assay agreed well with expected values for aminopropyltriethoxysilane grafted onto commercial silica. The assay also distinguished between reactive amines and protonated or poisoned surface amines on co-condensed SBA-15 materials. 

Note that we introduced the superscript CD in order to distinguish the expressions obtained by Clark and Davidson from those by Mayer, which will be given in the following marked by Ma. In a similar fashion, Mayer s partitioning of the total spin expectation value can be derived. Starting from Lowdin s expression for the total spin expectation value, Eq. (96), a one-electron basis set is introduced as in Eq. (102) and the numbers of a- and / -electrons, Na and N13, respectively, are replaced by sums over diagonal matrix elements Y (P"S)W and E (P S) w [cf. Eq. (104)], M... 

For over a century it has been known that two classes of variables have to be distinguished the microscopic variables, which are functions of the points of ClN and thus pertain to the detailed positions and motions of the molecules and the macroscopic variables, observable by operating on matter in bulk, exemplified by the temperature, pressure, density, hydro-dynamic velocity, thermal and viscous coefficients, etc. And it has been known for an equally long time that the latter quantities, which form the subject of phenomenological thermo- and hydrodynamics, are definable either in terms of expected values based on the probability density or as gross parameters in the Hamiltonian. But at once three difficulties of principle have been encountered. 

In practice decision makers typically are risk averse and the expected value approach does not take into account the variability of the solutions obtained under the probability distributions or scenarios considered for the uncertain parameters. Rosenhead et al. (1972) introduced the aspect of robustness as a criterion for strategic planning to address this issue. Building on the notion of robustness, Mulvey et al. (1995) developed the concept of robust optimization distinguishing between two different types of robust models. A model is solution robust if the solution obtained remains close to optimality for any realization of the uncertain parameters. The model itself is robust if it remains (almost) feasible for any realization of the uncertain parameters (model robust).36 Here, only solution robustness is of interest as the most important elements of uncertainty in production network design, namely demand volumes, costs, prices and exchange rates, should not lead to infeasibility problems under different scenarios considered. 

In general, a minimum of the energy surface corresponds to a set of stationary vibrational states of the molecular system. The position of the energy minimum is commonly called the equilibrium geometry Re. Analogously, we denote the expectation values for the molecular geometries in the vibrational states 0,1,2,... by R0, Rx, R2, etc. In most cases Ro is very close to Re. There are also exceptions to this correspondence which are important in the theory of intermolecular forces. We distinguish several cases ... 

So the expected value of the Guideline is in the field of comparison. Two types of comparison can be distinguished ... 

The formal definition of the NDF given in Eq. (4.11) is mathematically consistent, but difficult to implement in practice. It is therefore useful to define methods for estimating the NDF from a single realization of the granular flow. Note that mathematically a statistical estimate is a random variable, and thus should not be confused with the NDF, which is deterministic. In order to distinguish the estimated NDF from n, we will denote the estimate by h. Thus, for example, if the estimate is unbiased then (n) = n, where the expected value is taken with respect to the multi-particle joint PDF / defined in Eq. (4.7). 

It has been established in Sect.6 that whenever it becomes necessary to employ a mixed (rather than pure) ensemble, with density operator (44), the interatomic spin-coupling density Q ri,r2) is everywhere zero and thus yields zero for the expectation value of the spin scalar product Sx Sb there can be no spin coupling between A and B. Two regimes may thus be distinguished ... 

L(t) will denote the load, or number of jobs in the system, waiting or being served, at time t 0. For each model considered in this section, the distribution of L(t) approaches a limit as t — and the limit is referred to as the steady-state distribution of the number in the system. (As we said in Subsection 3.3, we do not distinguish carefully between steady-state and Kmiting distributions.) We denote by L a random variable whose distribution is this steady-state distribution. Let W denote the time-in-system or flow time of the nth arrival to the system. Then, for the models considered here, the distribution of W also has a limit as n —> , which we call the steady-state flow-time distribution. We denote by W a random variable with this distribution. In the same way we can define a random variable Wg whose distribution is the steady-state distribution of waiting time. Flow time and waiting time differ only in that the latter does not include service time. Most of the results given here concern the expected values of the random variables L, W, and Wg for various queues. These... 

Note that in various fields of science, when people talk about averaging a physical quantity, they usually mean calculating the expected value of that quantity (considered but not necessarily declared as a random variable). The truth is revealed by the fact that they use formulas like Eqs. (9.1) and (9.4) for those calculations. Also, the mass function and the density function are often not distinguished, but they are referred to by the same expressions probability density or differential distribution function and sometimes, rather loosely, as distribution function or just distribution. ... 

The expectation value D of the atomic dipole moment for our two-level system, which should be distinguished from the dipole matrix element, is... 

As is well known in nonequilibrium statistical mechanics, it is necessary to distinguish between energy-transfer processes and phase-relaxation processes in the description of the dynamics of complex systems. To monitor the coherence of vibrational motion, we may consider the expectation values of the position and momentum operators... 

It is important to distinguish between parameters of statistical distributions such as density function, expected value, and standard deviation on the one hand, and corresponding sample characteristics such as histograms, mean value and empirical standard deviation on the other. The latter are calculated from actual samples and will converge — in a stochastic sense — with increasing sample size towards the distribution parameters which are the usually unknown characteristics of the abstract probability distribution behind infinitely many samples of one and the same experiment. The sample-based quantities are also called estimates of the corresponding statistical parameters. 

The expectation value Dab of the dipole matrix element for our two-level system should be distinguished from the expectation value of the dipole moment in a specific state ir)... 

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