Notation-variable

The expressions Eqs. (2), (4) are completely general. To address the aspects important for the TMCs modelling, i.e. the energies of the corresponding electronic states, we notice that the statement that the motion of electrons is correlated can be given an exact sense only with use of the two-electron density matrix Eq. (4). Generally, it looks like [35] (with subscripts and variables notations w omitted for brevity) ... 

In terms of the state variable notation of Eq. (1), there are three states, three parameters, and four inputs (often the feedstream compositions of components B and C will be zero) ... 

The output vector, y, is of dimension m, which represents the linear combination of the states which are directly measurable. With state variable notation, one can achieve dynamic... 

Equations 8.7-4 also provide a means of identifying the equilibrium state when chemical reactions occur. To see this, consider first the case of a single chemical reaction occurring in a single phase (both of these restrictions will be removed shortly) in a closed system at constant temperature and pressure.The total Gibbs energy for this system, using the reaction variable notation introduced in Sec. 8.3, is... 

Kinetics variable notation is quite different from discipline to discipline and often inconsistent within a single discipline. In this book, r,- refers to the change in the number of moles of species i in a system per unit time (mol/ sec), Ri refers to the change in concentration of species i per unit time (molal/ sec), and /, refers to the flux of species across an interface (mol/m sec). 

The variable q, which is conjugate to the arc length position variable s, labels the normal mode. If equation (148) is written in discretized (bead variable) notation, q is actually Inp/L where... 

Note that in this equation, the complex variable notation has been changed from the "fused in most physics oriented works to the "/"commonly used in engineering work. This is in anticipation of using previously developed computer code which used the engineering notation. Some authors also note that the normalized equation is equivalent to working in a system of units in which h is equal to 1 and m is equal to 1/2. 

In matrix notation PCA approximates the data matrix X, which has n objects and m variables, by two smaller matrices the scores matrix T (n objects and d variables) and the loadings matrix P (d objects and m variables), where X = TPT... 

The notation < i j k 1> introduced above gives the two-electron integrals for the g(r,r ) operator in the so-called Dirac notation, in which the i and k indices label the spin-orbitals that refer to the coordinates r and the j and 1 indices label the spin-orbitals referring to coordinates r. The r and r denote r,0,( ),a and r, 0, ( ), a (with a and a being the a or P spin functions). The fact that r and r are integrated and hence represent dummy variables introduces index permutational symmetry into this list of integrals. For example,... 

The fact that the ratios of rates were much greater in chlorination than in nitration, prompted Dewar to suggest that the actual transition state was intermediate between the Wheland model and the isolated molecule model. He accommodated this variation in the relative rates within his discussion by treating yS as a variable whose value depended on the nature of the reaction. With the notation that y ) is the... 

In cement chemists notation, where subscripts x andj indicate variable unknown quantities. 

On page 235-241 is the explicit solution used in Excel format to make studies, or mathematical experiments, of any desired and possible nature. The same organization is used here as in previous Excel applications. Column A is the name of the variable, the same as in the FORTRAN program. Column B is the corresponding notation and Column C is the calculation scheme. This holds until line 24. From line 27 the intermediate calculation steps are in coded form. This agrees with the notation used toward the end of the FORTRAN listing. An exception is at the A, B, and C constants for the final quadratic equation. The expression for B was too long that we had to cut it in two. Therefore, after the expression for A, another forD is included that is then included in B. 

The variables F, K, D, r and d are all assumed to be random in nature following the Normal distribution, with the parameters shown in common notational form ... 

The random nature of most physieal properties, sueh as dimensions, strength and loads, is well known to statistieians. Engineers too are familiar with the typieal appearanee of sets of tensile strength data in whieh most of the individuals eongregate around mid-range and fewer further out to either side. Statistieians use the mean to identify the loeation of a set of data on the seale of measurement and the variance (or standard deviation) to measure the dispersion about the mean. In a variable x , the symbols used to represent the mean are /i and i for a population and sample respeetively. The symbol for varianee is V. The symbols for standard deviation are cr and. V respeetively, although a is often used for both. In this book we will always use the notation /i for mean and cr for the standard deviation. 

After defining fundamental terms used in probability and introducing set notation for events, we consider probability theorems facilitating tlie calculation of the probabilities of complex events. Conditional probability and tlie concept of independence lead to Bayes theorem and tlie means it provides for revision of probabilities on tlie basis of additional evidence. Random variables, llicir probability distributions, and expected values provide tlie means... 

So far, I have ignored the existenee of spin. Spin is an internal angular momentum that some partieles have and others do not. Eleetron spin is a two-valued quantity vve denote the spin variable for a single eleetron s, and the spin states are written o (s) and 3(s), or just a and p for short when the meaning is obvious. The notation I am going to use is that afsj) means eleetron 1 in spin state a. With an eye to the discussion above about indistinguishability, we consider the following four combinations of spin states for two electrons ... 

Don t confuse this with my earlier use of x for a space-spin variable the notation is common usage in both applications.) The Klein-Gordon equation is therefore... 

The solutions for the unperturbed Hamilton operator from a complete set (since Ho is hermitian) which can be chosen to be orthonormal, and A is a (variable) parameter determining the strength of the perturbation. At present we will only consider cases where the perturbation is time-independent, and the reference wave function is non-degenerate. To keep the notation simple, we will furthermore only consider the lowest energy state. The perturbed Schrodinger equation is... 

The kurtosis, the fourth moment, is a measure of peakedness or flatness of a distribution. It has no common notation (k is used here) and is given for a continuous random variable by... 

Since the space and time variables are the same throughout the equation, the notation... 

Note carefully that the same random variable (function) may have many different distribution functions depending on the distribution function of the underlying function X(t). We will avoid confusion on this point by adopting the convention that, in any one problem, and unless an explicit statement to the contrary is made, all random variables are to be used in conjunction with a time function X(t) whose distribution function is to be the same in all expressions in which it appears. With this convention, the notation F is just as unambiguous as the more cumbersome notation so that we are free to make use of whichever seems more appropriate in a given situation. 

We conclude this section by introducing some notation and terminology that are quite useful in discussions involving joint distribution functions. The distribution function F of a random variable associated with time increments fnf m is defined to be the first-order distribution function of the derived time function Z(t) = + fn),... 

In this connection, we shall often abuse our notation somewhat by referring to FXZx Ts as the joint distribution function of the random variables X(t + rx) and X(t + r2) instead of employing the more precise but cumbersome language used at the beginning of this paragraph. In the same vein, the distribution function FXJn.rym will be referred to loosely as the joint distribution function of the random variables X(t + rj),- -, X(t + r ), Y(t + ri), -,Y t + r m). 

Let us consider a function Z that depends on two variables, X and Y, and signify this with the notation Z = f(X, Y). In addition to designating Z as a function, we may also refer to Z as the dependent variable, and X and Y as the independent variables. We can write a differential expression dZ that tells us the change in the dependent variable Z arising from small changes in the independent variables, dX and d Y. The result is... 

The notation is meant to suggest that the frequency is variable and depends on the propagator matrix elements. The following criteria have proved valuable in choosing the variable coefficients of eq. IV.5 (1) at low temperature, the VQRS reference should weight the region around the potential minimum most heavily, and (2) at high temperature, our approximation should approach the classical limit ... 

Because the independent variable values are fixed for the problem, we may simplify notation by looking at u as a function of the variable 0 From now on we will therefore write u(X, 0) as u(0). 

Each quantized property can be identified, or indexed, using a quantum number. These are integers that specify the values of the electron s quantized properties. Each electron in an atom has three quantum numbers that specify its three variable properties. A set of three quantum numbers is a shorthand notation that describes a particular energy,... 

In addition to having to assign state variables to the strings of the DDF, we also have to assign properties to the alphabet symbols. In our flowshop example, the alphabet symbols can be interpreted as batches to be executed with a series of processing times. Thus, if we use the notation, (jc), to denote the state of partial solution, x, then... 

Partial least squares regression (PLS). Partial least squares regression applies to the simultaneous analysis of two sets of variables on the same objects. It allows for the modeling of inter- and intra-block relationships from an X-block and Y-block of variables in terms of a lower-dimensional table of latent variables [4]. The main purpose of regression is to build a predictive model enabling the prediction of wanted characteristics (y) from measured spectra (X). In matrix notation we have the linear model with regression coefficients b ... 

The NSS concept corresponds to an operator attached to an arbitrary number of nested sums. In other words, a NSS represents a set of summation symbols where the number of them can be variable. In a general notation one can write a NSS as E (j=i,f,s,L) where the meaning of this convention corresponds to perform all the sums involved in the generation of all the possible values of the index vector j under the fulfillment of the set of logical expressions collected in the components of the vector L. The elements of the vector j have the following limits ... 

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