First-order term

The complexity analysis shows that the load is evenly balanced among processors and therefore we should expect speedup close to P and efficiency close to 100%. There are however few extra terms in the expression of the time complexity (first order terms in TV), that exist because of the need to compute the next available row in the force matrix. These row allocations can be computed ahead of time and this overhead can be minimized. This is done in the next algorithm. Note that, the communication complexity is the worst case for all interconnection topologies, since simple broadcast and gather on distributed memory parallel systems are assumed. 

Continuity equation corresponding to the first-order terms... 

Before elosing this ehapter, it is important to emphasize the eontext in whieh the transition rate expressions obtained here are most eommonly used. The perturbative approaeh used in the above development gives rise to various eontributions to the overall rate eoeffieient for transitions from an initial state i to a final state f, these eontributions inelude the eleetrie dipole, magnetie dipole, and eleetrie quadrupole first order terms as well eontributions arising from seeond (and higher) order terms in the perturbation solution. 

The terms vanish, and the first-order terms reduee to ... 

The spherical geometry assumed in the Stokes and Einstein derivations gives the highly symmetrical boundary conditions favored by theoreticians. For ellipsoids of revolution having an axial ratio a/b, friction factors have been derived by F. Perrin, and the coefficient of the first-order term in Eq. (9.9) has been derived by Simha. In both cases the calculated quantities increase as the axial ratio increases above unity. For spheres, a/b = 1. 

The first-order term in this expansion renormalizes the potential V Q) while the bilinear term is analogous to the last term in (5.38). This is the linear-response theory for the bath. In fact, it shows... 

This treatment illustrates several important aspects of relaxation kinetics. One of these is that the method is applicable to equilibrium systems. Another is that we can always generate a first-order relaxation process by adopting the linearization approximation. This condition usually requires that the perturbation be small (in the sense that higher-order terms be negligible relative to the first-order term). The relaxation time is a function of rate constants and, often, concentrations. 

Note that in this case the spin-orbit coupling is included already in zero order. Including the first-order term from an expansion of K defines the Eirst-Order Regular Approximation (FORA) method. 

The first order term is known as the Wigner correction. ... 

Recall that equations 9.86 and 9.100 have been both derived using only the first-order terms in the Taylor series expansion of our basic kinetic equation (equation 9.77). It is easy to show that if instead all terms through second-order in 6x and 6t are retained, the continuity equation ( 9.86) remains invariant but the momentum equation ( 9.100) requires correction terms [wolf86c]. The LHS of equation 9.100, to second order in (ia (5 << 1, is given by... 

To the same order of approximation of the equations, that is, with only terms linear in / (v) kept, better approximations to the viscosity may be found by considering the equations of higher order than Eqs. (1-86) and (1-87). These new equations will, to this order of approximation, have zero on the left sides (since the higher order coefficients are taken equal to zero) on the right sides appears the factor (p/fii) multiplied by a series of terms like those in Eq. (1-110). Using these equations, and the first order terms of Eq. (1-86) for arbitrary v,... 

We may solve for the electron distribution function by expanding it in Legendre polynomials in cos 6 (where v = (v,6,Fourier series in cot we shall use here only the first-order terms ... 

To order e2 the Coulomb interaction term contributes to the first-order term in the 5-matrix expansion, i.e., to fd4xJffin(x), and the term contributes to second-order. Diagrammatically, we have illustrated these contributions in Fig. 11-1 ( stands for Coulomb, / for transverse photons). To the order indicated, the part of the 5-matrix contributing to the process is... 

If the initial ground-state wavefunction (/(q is nondegenerate, the first-order term (i. e., the second term) in Eq. (1) is nonzero only for the totally-symmetrical nuclear displacements (note that g, and (dH/dQi) have the same symmetry). Information about the equilibrium nuclear configuration after the symmetrical first-order deformation will be given by equating the first-order term to zero. 

In the case of the hexacarbonyls, the rate-expression contains not only the same type of first-order term but in addition one second-order overall. For good entering groups (but not CO, for example) the rate expression contains a term strictly first-order in both the complex and the entering nucleophile. The first-order rates of CO exchange are practically identical with the rates of substitution in hydrocarbon solvents, but there is nevertheless some acceleration in ether (THF, dioxan) solutions. This solvent-dependence is not so well-marked ° as in the case of nickel tetracarbonyl. The second-order rate of substitution very strongly depends upon the basicity of the entering nucleophile... 

A comparison of the V(V) oxidations of acetoin, CH3CH(OH)COCH3, and 3-hydroxy-3-methylbutan-2-one, (CH3)2C(OH)COCH3, shows that whilst both rate laws include first-order terms in substrate and oxidant, the acidity dependence for the former compound is purely ho but that for the latter is a+bho). The C-methyl compound consumes only 2 equivalents of V(V) to give acetone and a mechanism similar to that for the oxidation of pinacol is proposed , viz. 

With the choice a = 0, the total eigenfunction xp io first order is normalized. To show this, we form the scalar product xp xp ) using equation (9.29) and retain only zero-order and first-order terms to obtain... 

Again, let us assume that an estimate kw of the unknown parameters is available at the j,h iteration. Linearization of the output vector around and retaining first order terms yields... 

It is cumbersome to write the partial fraction with complex numbers. With complex conjugate poles, we commonly combine the two first order terms into a second order term. With notations that we will introduce formally in Chapter 3, we can write the second order term as... 

In establishing the relationship between time-domain and Laplace-domain, we use only first and second order differential equations. That s because we are working strictly with linearized systems. As we have seen in partial fraction expansion, any function can be "broken up" into first order terms. Terms of complex roots can be combined together to form a second order term. 

To complete the first order terms, the exchange-repulsion energy can be evaluated through an overlap model [14, 59] as ... 

Hv>q—Ho,o) in Eq.(3.9) can usually not be small in excited configuration terms, whereas in transferred configuration terms it can be. Even a first-order term of the form... 

In the case of degeneracy where one of the monotransferred configurations happens to have the same energy as the initial configuration, the first-order term of Eq. (3.10) appears. Obviously, such a case is possible only in regard to the transfer of one electron from HO MO of the donor molecule to LU MO of the acceptor molecule. 

The first term in both Equations 17 and 18 is the constant surface-tension contribution and the second term gives the first-order contribution resulting from the presence of a soluble surfactant with finite sorption kinetics. A linear dependence on the surfactant elasticity number arises because only the first-order term in the regular perturbation expansion has been evaluated. The thin film thickness deviates negatively by only one percent from the constant-tension solution when E = 1, whereas the pressure drop across the bubble is significantly greater than the constant-tension value when E - 1. 

In a perturbation theory treatment of the total (not just electrostatic) interaction between the molecule and the point charge, QV(r) is the first-order term in the expression for the total interaction energy (which would include polarization and other effects). 

The first-order term Ai arises if two partially occupied interacting orbitals are degenerate or nearly degenerate in energy. The second-... 

If the sample is spun exactly at fjRL = 54.736° and at infinite spinning speed, the first-order term /, vanishes (33) and the second-order anisotropic term yields... 

Figure 17.1 shows the calculation results. The mass of CrVI decreases at a rate mirroring the increase in Cr111 mass, which is twice the rate at which Reaction 17.28 proceeds. Dissolved sulfide in the simulation is divided approximately evenly between HS- and H2S(aq), since pH is held to 7. The reaction consumes H2S(aq) as well as Cr()/, causing the concentration of each to decline. Since the two concentrations appear as first order terms in the rate law, reaction rate also decreases with time. 

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