Mechanism quadratic

Nonlinear response because of epigenetic mechanism (quadratic relation used as representative)... 

Hach molecular mechanics method has its own functional form MM+. AMBER, OPL.S, and BIO+. The functional form describes the an alytic form of each of th e term s in th e poteri tial. For exam pie, MM+h as both a quadratic and a cubic stretch term in th e poten tial whereas AMBER, OPES, and BIO+ have only c nadratic stretch term s, I h e functional form is referred to here as the force field. For exam pie, th e fun ction al form of a qu adratic stretch with force constant K, and equilibrium distance i q is ... 

A restrain t (not to be confused with a Model Builder constraint) is a nser-specified one-atom tether, two-atom stretch, three-atom bend, or four-atom torsional interaction to add to the list ol molec-11 lar mechanics m teraction s computed for a molecule. These added iiueraciious are treated no differently IVoin any other stretch, bend, or torsion, except that they employ a quadratic functional form. They replace no in teraction, on ly add to the computed in teraction s. 

This formula is exact for a quadratic function, but for real problems a line search may be desirable. This line search is performed along the vector — x. . It may not be necessary to locate the minimum in the direction of the line search very accurately, at the expense of a few more steps of the quasi-Newton algorithm. For quantum mechanics calculations the additional energy evaluations required by the line search may prove more expensive than using the more approximate approach. An effective compromise is to fit a function to the energy and gradient at the current point x/t and at the point X/ +i and determine the minimum in the fitted function. 

This equation is a quadratic and has two roots. For quantum mechanical reasons, we are interested only in the lower root. By inspection, x = 0 leads to a large number on the left of Eq. (1-10). Letting x = leads to a smaller number on the left of Eq. (1-10), but it is still greater than zero. Evidently, increasing a approaches a solution of Eq. (1-10), that is, a value of a for which both sides are equal. By systematically increasing a beyond 1, we will approach one of the roots of the secular matrix. Negative values of x cause the left side of Eq. (1-10) to increase without limit hence the root we are approaching must be the lower root. 

PW91 (Perdew, Wang 1991) a gradient corrected DFT method QCI (quadratic conhguration interaction) a correlated ah initio method QMC (quantum Monte Carlo) an explicitly correlated ah initio method QM/MM a technique in which orbital-based calculations and molecular mechanics calculations are combined into one calculation QSAR (quantitative structure-activity relationship) a technique for computing chemical properties, particularly as applied to biological activity QSPR (quantitative structure-property relationship) a technique for computing chemical properties... 

Electrostrictive materials are materials that exhibit a quadratic relationship between mechanical stress and the square of the electric polari2ation (14,15). Electrostriction can occur in any material. Whenever an electric field is appHed, the induced charges attract each other, thus, causing a compressive force. This attraction is independent of the sign of the electric field and can be approximated by... 

For each EA spectrum, the transmission T was measured with the mechanical chopper in place and the electric field off. The differential transmission AT was subsequently measured without the chopper, with the electric field on, and with the lock-in amplifier set to detect signals at twice the electric-field modulation frequency. The 2/ dependency of the EA signal is due to the quadratic nature of EA in materials with definite parity. AT was then normalized to AT/T, which was free of the spectral response function. To a good approximation [18], the EA signal is related to the imaginary part of the optical third-order susceptibility ... 

Most types of SS used for water treatment have an Austenitic crystalline structure (centre-faced cubic). Others are Ferritic (centred cubic), Marstenitic (quadratic), or Austeno-ferritic types, which have superior mechanical strength and are resistant to stress corrosion. 

Figure 37. Electronic excitation of the NaK wavepacket from the inner turning point of the ground X state. The X A transition is considered. The initial wave packet is prepared by two quadratically chirped pulses within the pump-dump mechanism. Taken from Ref. [37].

The reciprocals of the time constants, x, and x2, are the rate constants k, and k2. The weights of the exponentials (ii and w2) are complicated functions of the transition rates in Eq. (6.25). Flowever, the rate constants are eigenvalues found by solving the system of differential equations that describe the above mechanism. A, and k2 are the two solutions of the quadratic equation ... 

If A is a square matrix and AT is a column matrix, the product AX is a so a column. Therefore, the product XAX is a number. This matrix expression, which is known as a quadratic form, arises often in both classical and quantum mechanics (Section 7.13). In the particular case in which A is Hermitian, the product XxAX is called a Hermitian form, where the elements of X may now be complex. 

As shown by Strobl [230], the integral breadths B in a series of reflections is increasing quadratically if (1) the structure evolution mechanism leads to a convolution polynomial, (2) the polydispersity remains moderate, (3) the rod-length distributions can be modeled by Gaussians (cf. Fig. 8.44). For the integral breadth it follows... 

Burch (1983) suggests that repair mechanisms cause a non neglectible complication for extrapolation from high to low doses and presents a modification of the linear-quadratic formula given above. Katz and Hofmann (1982) carried out an analysis of particle tracks with the result that they find no basis for a linear or linear-quadratic extrapolation to low doses. Van Bekkum and Bentvelzen (1982) present a hypothesis of the gene transfer-... 

Both solvent exchange reactions show an interchange (I) mechanism (Figs. 25 and 26). Therefore, the quadratic pyramidal ground state with a second solvent molecule in the second... 

Linear chain termination is not, however, a necessary condition for the critical behavior. Indeed, with mechanisms V and XII, chain termination is quadratic (v v,172), but critical transition does take place because hydroperoxide decomposes into radicals that contribute to chain propagation. As a result, v (v [ROOH])1/2 v, [ROOH]172, and v [ROOH] (see Equation (14.11)) which explains the critical behavior. 

The mechanical modes whereby molecules may absorb and store energy are described by quadratic terms. For translational kinetic energy it involves the square of the linear momentum (E = p2/2m), for rotational motion it is the square of angular momentum (E = L2121) and for vibrating bodies there are both kinetic and potential energy (kx2/2) terms. The equipartition principle states that the total energy of a molecule is evenly distributed over all available quadratic modes. 

From point (1), the velocity profile is parabolic that is, the linear (axial) velocity u depends quadratically on radial position r, as described by fluid mechanics (see, e.g., Kay and Nedderman, 1974, pp. 69-71) ... 

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