Notation simplifying

Generally, a set of coupled diffusion equations arises for multiple-component diffusion when N > 3. The least complicated case is for ternary (N = 3) systems that have two independent concentrations (or fluxes) and a 2 x 2 matrix of interdiffusivities. A matrix and vector notation simplifies the general case. Below, the equations are developed for the ternary case along with a parallel development using compact notation for the more extended general case. Many characteristic features of general multicomponent diffusion can be illustrated through specific solutions of the ternary case. 

This notation simplifies considerably the mathematical expressions, and there is no need to use complicated trigonometric formulae when dealing with the changes in the phase and ampUtude of AC signals. 

C. How does noble-gas notation simplify writing an atom s electron configuration ... 

SMILES notation (Simplified Molecular Input Line Entry System) used in the Pomona College MedChem project ( ) the Dyson notation (5) adopted as a standard by the International Union for Pure and Applied Chemistry (lUPAC) and Wiswesser Line Notation (WLN), which became the actual standard used in the chemical industry (6). The rules of WLN were standardized by the Chemical Notation Association (CNA), but the two major WLN data bases. Index Chemicus Registry System, ICRS, from the Institute for Scientific Information and the Commercially Available Organic Chemicals Index, CAOCI (7) used somewhat different conventions. 

In this equation, 01 is the ifeqiieney of the RF irradiation, oiq is the Lannor ifeqiieney of the spin, is the spm-spm relaxation time andM is the z magnetization of the spin system. The notation ean be simplified somewhat by defining a eomplex magnetization, AY, as in equation (B2.4.3). 

We shall assume, for simplifying the notation, that the k values are positive. For a phase-inverting reaction, the wave function of the transition state is therefore written as... 

In what follows, the subscript M will be omitted to simplify the notations. If the initial point is P po, qa) and we are interested in deriving the value of A(= Am) at a final point Q p, q) then one integral equation to be solved is... 

The Wiswesser Line Notation (WLN) was introduced in 1946, in order to organize and to systematically describe the cornucopia of compounds in a more concise manner. A line notation represents a chemical structure by an alphanumeric sequence, which significantly simplifies the processing by the computer [9-11], (n many cases the WLN uses the standard symbols for the chemical elements. Additionally, functional groups, ring systems, positions of ring substituents, and posi-... 

In 1986, David Weininger created the SMILES Simplified Molecular Input Line Entry System) notation at the US Environmental Research Laboratory, USEPA, Duluth, MN, for chemical data processing. The chemical structure information is highly compressed and simplified in this notation. The flexible, easy to learn language describes chemical structures as a line notation [20, 21]. The SMILES language has found widespread distribution as a universal chemical nomenclature... 

A. similar expression can also be introduced for Ab- With this notation F simplifies to ... 

The electron configuration is the orbital description of the locations of the electrons in an unexcited atom. Using principles of physics, chemists can predict how atoms will react based upon the electron configuration. They can predict properties such as stability, boiling point, and conductivity. Typically, only the outermost electron shells matter in chemistry, so we truncate the inner electron shell notation by replacing the long-hand orbital description with the symbol for a noble gas in brackets. This method of notation vastly simplifies the description for large molecules. 

Molar concentrations are used so frequently that a symbolic notation is often used to simplify its expression in equations and writing. The use of square brackets around a species indicates that we are referring to that species molar concentration. Thus, [Na ] is read as the molar concentration of sodium ions. ... 

The kinetic chain length u may also be viewed as merely a cluster of kinetic constants and concentrations which was introduced into Eq. (6.54) to simplify the notation. As an alternative, suppose we define for the purposes of this chapter a fraction p such that... 

This notation is simplified still further by defining the ratio of constants... 

Although there are a total of four different rate constants for propagation, Eq. (7.12) shows that the relationship between the relative amounts of the two monomers incorporated into the polymer and the composition of the monomer feedstock involves only two ratios of different pairs of these constants. Accordingly, we simplify the notation by defining... 

Those involving volume approximations. We replace njVi by Vj and identify the latter as the total volume of the solution. For dilute solutions these do not introduce serious errors and do simplify the notation somewhat. 

In the sequel we consider different functional spaces. To simplify the notation we write H Q) instead of [77 (12)] and so on. 

To simplify the notations we do not indicate the dependence of the solutions on the parameters s, 5. Our aim is first to prove the existence of solutions to (5.185)-(5.188) and next to justify the passage to limits as c, 5 —> 0. A priori estimates uniform with respect to s, 5 are needed to study the passage to the limits, and we shall derive all the necessary inequalities while the existence theorem is proved. 

In the analysis of composites it is convenient to use matrix notation because this simplifies the computations very considerably. Thus we may write the above equations as (see Appendix E)... 

A fourth-order tensor can be written as a 9x9 array in analogy to Equation (A.51) but, by use of contracted notation, is sometimes drastically simplified to a 6 x 6 symmetric array. 

Bom and Oppenheimer tackled the problem quantum-mechanically in 1927 their treatment is pretty involved, but the basic physical idea is as outlined above. To simplify the notation, I will write the total Hamiltonian as follows ... 

For large molecules, very many terms contribute to the electronic Hamiltonian. To simplify the notation, I am going to collect together all those terms that depend explicitly on the coordinates of a single electron and write them as... 

I have grouped the terms on the right-hand side together for a reason. We normally simplify the notation along the lines discussed for dihydrogen in Chapter 4, and write the electronic Hamiltonian as a sum of the one-electron and two-electron operators already discussed. 

Taking a product over all sites and simplifying notation somewhat, we get that... 

Numbers such as these are very awkward to work with. For example, neither of the numbers just written could be entered directly on a calculator. Operations involving very large or very small numbers can be simplified by using exponential (scientific) notation. To express a number in exponential notation, write it in the form... 

A major advantage of exponential notation is that it simplifies the processes of multiplication and division. To multiply, add exponents ... 

One of the uses of the joint characteristic function is in the calculation of joint moments. In order to simplify the notation, we shall only... 

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). 

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