Algebraic equivalent

It has been pointed out [138] that algebraically equivalent expressions can be derived without invoking a surface solution model. Instead, surface excess as defined by the procedure of Gibbs is used, the dividing surface always being located so that the sum of the surface excess quantities equals a given constant value. This last is conveniently taken to be the maximum value of F. A somewhat related treatment was made by Handa and Mukeijee for the surface tension of mixtures of fluorocarbons and hydrocarbons [139]. 

An equation algebraically equivalent to Eq. XI-4 results if instead of site adsorption the surface region is regarded as an interfacial solution phase, much as in the treatment in Section III-7C. The condition is now that the (constant) volume of the interfacial solution is i = V + JV2V2, where V and Vi are the molar volumes of the solvent and solute, respectively. If the activities of the two components in the interfacial phase are replaced by the volume fractions, the result is... 

Equation (10) represents the simplest form of the isokinetic relationship however, several equivalent expressions are also possible and will now be discussed and shown in diagrams. It should be commented in advance that algebraic equivalence does not imply equivalence from the statistical point of view (see Section IV.). 

Algebraically equivalent formulation as used in pocket calculators (beware of numerical artifacts when using Eqs. (2.7-2.9) cf. Table 1.1) ... 

Equation 59-12, of course, is simply the algebraic equivalent of the equation found above.]... 

Figure 1. Diagrammatic representation of the terms that contribute to the irreducible one-particle BriUouin conditions, Eq. (167), with their algebraic equivalents. The last line represents the commutators (i.e., the difference of the above values).

Accdg to von Stein Alster (Ref 41), accurate determination of isochoric adiabatic flame temp of an expl often involves a series of tedious calcns of the equilibrium of compn of the expln products at several temps. Calcg the expln product compn at equilibrium is a tedious process for it.requires the soln of a number of non-linear simultaneous equations by a laborious iterative procedure. Damkoehler Edse (Ref 11) developed a graphical procedure and Wintemitz (Ref 26a) improved it. by transforming it into its algebraic equivalent. Unfortunately both.methods proved less useful with.hetorogeneous equilibria which. contain solid carbon... 

The first two chapters are about moduli of abelian varieties, i.e. about classifying abelian varieties. An abelian variety is a proper variety endowed with a group structure. It turns out that any abelian variety A is projective there exists some ample line bundle on A. An ample line bundle determines an isogeny A A — A, which depends on C only up to algebraic equivalence. Such a morphism A is called a polarization, it is called a principal polarization if A is an isomorphism. 

Note that isotope i is usually that of high abundance in the sample and k the isotope of high abundance in the spike, but using the reverse definition is algebraically equivalent. Note that the equation simplifies if the spike isotope is not present in the sample. 

Algebraically equivalent forms may be used in place of 103K/r, such as kK/T or 103(7yK) 1. 

One can verify directly that this is a homomorphism GLj - GL3. Obviously it contains information specifically about quadratic forms as well as about GL2—the orbits are isometry classes. We will touch on this again when we mention invariant theory in (16.4), but for now we use representations merely as a tool for deriving structural information about group schemes. The first step is to use a Yoneda-type argument to find the Hopf-algebra equivalent. 

Very recently Sinha et. al /13 6/ have analytically shown that the CC—based LRT for IP is algebraically equivalent to the CC formalism of Haque and Mukherjee/69/ for one-hole valence problem. This indicates that the CC equations for determining the IP s can be cast into an equivalent eigenvalue equation. In fact Sinha et. aj /136/ have demonstrated that the relation between the CC-LRT for IP and the errresponding CC theory is the same as that between a Cl problem and the associated partitioned problem as obtained by the Soliverez transformation/137/. For open-shells containing more than one valence, the correspondence between the CC-LRT and the CC equations no longer holds, but Sinha et. al showed the CC... 

This quantity has the property of additivity so that estimates from several sources of random error may be combined, and it is useful in probability calculations. An alternative form of S, algebraically equivalent and often more convenient for use with a calculator or computer, is... 

What you are doing with this procedure is the algebraic equivalent of drawing a straight line between the two points [.r , / ] and [xp, fp] on a plot of/ versus x and using the intersection of this line with the x axis as the next estimate of the root. 

These are known as the Roothaan equations. They represent an algebraic equivalent to the Hartree-Fock equations. The approximate eigenvalues represent orbital energies. By Koopmans theorem, — approximates the ionization energy for an electron occupying orbital a. The orbital energies can be determined directly Ifom the n roots of the secular equation... 

The analogous expansion may be developed for the nth-order perturbed wave function P(n) in Eq. (22b), but they must be expressed in terms of open diagrams, i.e., those in which all the lines are not closed into loops. Strictly speaking, such diagrams occur in the expansion of the wave operator, a concept studied at length by Lowdin,35 since their algebraic equivalents are second-quantized operators. The nth-order wave... 

Algebraically equivalent fonns may be used in place of 10 K/r, such as kK/T or 10 (r/K)". From the following equation one can see by writing the first few terms of a series expansion... 

The majority of quantum-chemistry calculations have been carried out by employing the independent particle model in the framework of the HF method. In the most widely used approach molecular orbitals are expanded in predefined one-particle basis functions which results in recasting the integro-differential HF equations into their algebraic equivalents. In practice, however, the basis set used is never complete and very often far too limited to describe essential features of HF orbitals, for example, their behaviour in the vicinity of nuclei. That is why such calculations always suffer from the so called basis set truncation error . This error is difficult to estimate and often leads to low credibility of the results. 

To start, we note that the short time Kst(T ) and Grote-Hynes kgh(T ) transmission coefficients are algebraically equivalent [23]. However, Kst(T) and Kgh(T ) are useful expressions in different physical regimes. Eqs. (3.50) and (3.51) for Kst(T) provide a useful parameterization of k(T ) only for reactions for which the rate constant k T) is determined by short time dynamics while Eqs. (3.46) and (3.47) provide a useful parameterization only for reactions for which k T) is determined by slow variable dynamics. Nearly equivalently, Eqs. (3.50) and (3.51) apply to sharp barrier reactions, where the sharp barrier limit is defined as comip oc while Eqs. (3.46) and (3.47) apply to flat barrier reactions, where the flat barrier limit is defined as (Ormf 0. (The sharp barrier limit is taken as comip oo, not as PMF oc as in Section III.B, isasmuch as sharp barrier reactions are short time, high-frequency processes for which oomip is the physical barrier frequency. The converse argument yields the flat barrier limit as copmf 0.)... 

Because of their algebraic equivalence, Kgh(T ) and KsT(r) are numerically identical and thus agree equally well with MD results. Com-... 

Equation (14-24) and (14-25) are based on algebraic equivalents of infinite power series. For example. 

Algebraically equivalent models A set of functionally equivalent process models can be algebraically equivalent, when one can transform any member of the set to any other one using algebraic transformations. This however is not always the case. An algebraically transformed model is the same fi-om the process engineering point of view. Such algebraically equivalent models form a model class. 

Both functionally and algebraically equivalent models The above properties of equivalent state-space models raises the question What are the conditions of functionally equivalent process models that make them algebraically equivalent . It follows from the definitions that such models may be obtained in case of a modelling goal specified... 

It means that the above requirements are necessary but not sufficient conditions for functionally equivalent models being also algebraically equivalent. 

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