Difference formula

Third card FORMAT(8F10.2), size of increments to be used in central difference formula for calculating derivatives with respect to the independent variables. 

A finite difference formula is used to estimate the second derivatives of the coordinate vector with respect to time and S is now a function of all the intermediate coordinate sets. An optimal value of S can be found by a direct minimization, by multi-grid techniques, or by an annealing protocol [7]. We employed in the optimization analytical derivatives of S with respect to all the Xj-s. 

The leap-frog algorithm uses the simplest central difference formula for a derivative... 

In the CIELAB and CIELUV color spaces, the difference between a batch sample and a reference standard designated with a subscript s, can be designated by its components, eg, AAL = L — L. The three-dimensional total color differences are given by EucHdian geometry as the 1976 CIE lYa b and 1976 CIE lYu Y color difference formulas ... 

Color Difference Evaluation. Shade evaluation is comparable in importance to relative strength evaluation for dyes. This is of interest to both dye manufacturer and dye user for purposes of quaUty control. Objective evaluation of color differences is desirable because of the well-known variabihty of observers. A considerable number of color difference formulas that intend to transform the visually nonuniform International Commission on Illumination (CIE) tristimulus color space into a visually uniform space have been proposed over the years. Although many of them have proven to be of considerable practical value (Hunter Lab formula, Friele-MacAdam-Chickering (FMC) formula, Adams-Nickerson formula, etc), none has been found to be satisfactorily accurate for small color difference evaluation. Correlation coefficients for the correlation between average visually determined color difference values and those based on measurement and calculation with a formula are typically of a magnitude of approximately 0.7 or below. In the interest of uniformity of international usage, the CIE has proposed two color difference formulas (CIELAB and CIELUV) one of which (CIELAB) is particularly suitable for appHcation on textiles (see Color). 

The second class of multivariable optimization techniques in principle requires the use of partial derivatives, although finite difference formulas can be substituted for derivatives such techniques are called indirect methods and include the following classes ... 

Logarithms 21. Binomial Theorem 22. Progressions 23. Summation of Series by Difference Formulas 23. Sums of the first n Natural Numbers 24. Solution of Equations in One Unknown 24. Solutions of Systems of Simultaneous Equations 25. Determinants 26. 

There have been at least six different formulas proposed and used to determine the proper and adequate size of rupture disk openings for a specific relieving condition. The earlier studies of Sylvander and Katz [25] led to the development of the ASME and API recommendations. This approach assumes that a fire exists under or around the various vessels in a process. This fire may have... 

Also introduced was sodium metaphosphate [Hagan phosphate or Calgon , represented at the time by the formula (NaP03)6, although it tends to have a different formula today]. 

The formulas of ionic compounds have a different meaning from those of molecular compounds. Each crystal of sodium chloride has a different total number of cations and anions. We cannot simply specify the numbers of ions present as the formula of this ionic compound, because each crystal would have a different formula and the subscripts would be enormous numbers. However, the ratio of the number of cations to the number of anions is the same in all the crystals, and the chemical formula shows this ratio. In sodium chloride, there is one Na+ ion for each Cl ion so its formula is NaCl. Sodium chloride is an example of a binary ionic compound, a compound formed from the ions of two elements. Another binary compound, CaCl2, is formed from Ca2+ and Cl- ions in the ratio 1 2, which is required for electrical neutrality. 

Green s difference formulae. For the simplest operator Lu — u" the following identities are valid ... 

Green s difference formula admits an alternative form... 

Some difference formulae. In the sequel, when dealing with various difference expressions, we shall need the formulae for difference differentiating of a product, for summation by parts and difference Green s formulae. In this section we derive these formulae within the framework similar to the appropriate apparatus of the differential calculus. Similar expressions were obtained in Section 2 of Chapter 1 in studying second-order difference operators, but there other notations have been used. It performs no difficulty to establish a relationship between formulae from Section 2 of Chapter 1 and those of the present section. 

In Section 1 we have already introduced two types of difference derivatives for grid functions the left and the right ones, which correspond to different formulae for difference differentiating of a product... 

Via transform of the first summand in (46) by the Green difference formula (8") we find that... 

By appeal to the Green difference formula derived in Chapter 2, Section 3.1... 

The partial differential equations describing the catalyst particle are discretized with central finite difference formulae with respect to the spatial coordinate [50]. Typically, around 10-20 discretization points are enough for the particle. The ordinary differential equations (ODEs) created are solved with respect to time together with the ODEs of the bulk phase. Since the system is stiff, the computer code of Hindmarsh [51] is used as the ODE solver. In general, the simulations progressed without numerical problems. The final values of the rate constants, along with their temperature dependencies, can be obtained with nonlinear regression analysis. The differential equations were solved in situ with the backward... 

We have developed several examples of fruitful areas in this chemistry that demonstrate feasibility. As one might expect, a list of compounds related to the pnictates proposed in Table 14.4 as the trivalent pnictogens and trielides should yield compounds of similar chemical composition. However, we can think of these systems in a slightly different manner. We can imagine a different formula from that in the previous equations ... 

Deactivating catalyst 319 Dead zones 159, 162, 163 Degree of segregation 471 Density influences 492 Desorption of solute 578, 579 Difference differential equation 579 Difference formulae for partial differential equations 268 Differential column 167... 

Of course, it is apparent that KCfi is given by differing formulas in the work of CGM and Pecora. The formula of CGM correctly answers the question posed in this paper. However, as we shall see later, the formula of Pecora correctly answers quite a different, but related question. For now, we turn to the remaining paper which is to be considered. 

All of the equations developed in this section assume that payments are made at the beginning of a period and interest is calculated at the end of the period. The bank in Example 10-9 would probably ask for the interest at the time the payment is made. This is not uncommon. If this is the case, different formulas must be developed for the various annuities. 

This case was studied by Nahavandi and Catanzaro (Nl) using an explicit forward difference formula with Ax set equal to the pipe length and the time step dictated by stability considerations. 

We can understand how —d2fix)/dx2 shows us where fix) is locally concentrated or depleted in another way by approximating it by the finite difference formula ... 

Arts. They were discussing Sec. 8.1. Since the student got the answer correct despite hearing the wrong name, they must have been discussing the number of moles of reactants and products. The numbers of moles of HF and HC1 would be the same in the reaction, but since they have different formula weights, their masses would be different. 

The best packages for stiff equations (see below) use Gear s backward difference formulas. The formulas of various orders are [Gear, G. W., Numerical Initial Value Problems in Ordinary Differential Equations, Prentice-Hall, Englewood Cliffs, N.J. (1971)]... 

While quantum chemistry cannot be used to work with the Landauer/Buttiker/Imry formula as it stands, a very different approach based on nonequilibrium Green s functions yields a different formula (sometimes called the Caroli formula [68], or the NEGF formula in the Landauer/Imry/Buttiker limit). It is, for the current I ... 

These are not isomers, because they have different formulas C4H10 and C4H8. 

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