Interactions, linear

Consider an ensemble of harmonic oscillators interacting linearly with an ion of charge number z, so that the potential energy of the system is given by ... 

Properties of nondiagonal rotation-vibration interactions Linear molecules... 

Fig. 8. Summary of the structural characteristics of collagen IV, VI, and dogfish egg case collagen. These collagens can interact linearly end-to-end through their N- and G-termini (represented in green and purple, respecdvely). Dogfish egg case and...

However, this ceases to hold when Gd contains also the self-energy part due to interactions. Using an approximate solution [13] for Gd for an electron-phonon interaction linear in the boson coordinates, one finds [11] that to lowest-order in the electron-phonon coupling aq 2, Ipc = Ipc + AIpc, with... 

To draw conclusions about the obtained regression model it is necessary to interpret the outcomes or to translate the model into the researcher s language. Since a linear model is inadequate, one must when interpreting the model analyze the even the factor interactions. Linear regression coefficients show by their values approximately the same effect of all factors on response. Their signs are identical so that the response value increases with an increase in factor value. The response or coefficient of the separation value in the last trial reaches the value from the reference... 

Porter GJ, Hill DR (1996) Interactive linear algebra a laboratory course using Mathcad. Springer Verlag, New York... 

For particular lattices with two molecules per unit cell, with simple structure and nearest-neighbor interactions (linear alternating chains, square 2D lattices, or simple cubic lattices of the NaCl type), the absorption occurs for two values of q q = 0 (the b component) and q = Q (the a component) see Section II.B.l.b. In these simple lattices the environment of the site is symmetric and the complete inversion in P, TP, (2.77) is not necessary, because, in the coupling of P, to P, only the completely symmetric state S> intervenes ... 

One considers a particle interacting linearly with an environment constituted by an infinite number of independent harmonic oscillators in thermal equilibrium. The particle equation of motion, which can be derived exactly, takes the form of a generalized Langevin equation, in which the memory kernel and the correlation function of the random force are assigned well-defined microscopic expressions in terms of the bath operators. 

At the center of the echo all resonance offsets from interactions linear in the spin quantum number are canceled as long as these interactions operate for the full duration ofTE. Linear spin interactions include chemical shifts, heteronu-clear dipolar couplings, field inhomogeneity, field gradients, and transmitter frequency offsets but do not include quadru-polar and homonuclear dipolar couplings. There will however be a net phase evolution induced by an interaction to the extent its duration or intensity is not balanced with respect to the two halves of TE (that is, the balance with respect to amount of phase evolution on either side of the 180° pulse). 

In order to describe the state of the solvent, we represent it as a bath of harmonic oscillators, which interact linearly with the reactant. The corresponding Hamiltonian is written in the form ... 

This means that a particle of mass M interacts linearly with a chain of infinitely many particles with masses mj, and so on, which also interact with each other via a linear coupling (only nearest-neighbor interactions are considered). The dynamics of the particle of mass M is defined by its space coordinate x and velodty v. The space coordinates and velocities of the particles of mass m,- are denoted by the symbols y, and w, respectively. More proper variables are the relative distances... 

Zeeman interaction Linear with Hq, at high Hq high population difference Enery level splitting o 7 Hg (same effect in both phases) 50 MHz... 

Thus, in both cases, the molecular unit can be tailored to meet a specific requirement. A second crucial step in engineering a molecular structure for nonlinear applications is to optimize the crystal structure. For second-order effects, a noncentrosymmetrical geometry is essential. Anisotropic features, such as parallel conjugated chains, are also useful for third-order effects. An important factor in the optimization process is to shape the material for a specific device so as to enhance the nonlinear efficiency of a given structure. A thin-film geometry is normally preferred because nonlinear interactions, linear filtering, and transmission functions can be integrated into one precise monolithic structure. 

This beta function is zero in the absence of dipolar interactions. It vanishes for interactions linear in the spin operator and has nonzero values only for bilinear interactions. Furthermore, it is zero for Je approaching zero, so that a nonzero signal indicates residual anisotropic interactions, and it is free of signal attenuation by relaxation. The shape of the beta function has been shown to depend strongly on the strain in rubber samples [Cal3]. 

Several review articles deal extensively with the absorption and Raman spectroscopic manifestations of localised or resonance modes in impure crystals89 140 149). The experimental status is well documented in these and particularly in a recent compendium9). It is recalled that the emphasis of these reviews is different from that in this Report. They treat effects due to mass or force constant defects in the crystal. These are essentially averages over a vibrational period of the mode. In opposition, our concern is with effects coming from the phase-concerted interaction (linear in the vibrational coordinate). We turn to this in the next section, though for brevity s sake we shall not continue to carry the adjective phase-concerted in specifying the linear interaction. 

Other models have been proposed which follow the outlines of the equations already discussed. Equations with parameters that vary as a function of temperature, sunlight, and nutrient concentration have been presented by Davidson and Clymer (9) and simulated by Cole (10). A set of equations which model the population of phytoplankton, zooplankton, and a species of fish in a large lake have been presented by Parker (II). The application of the techniques of phytoplankton modeling to the problem of eutrophication in rivers and estuaries has been proposed by Chen (12). The interrelations between the nitrogen cycle and the phytoplankton population in the Potomac Estuary has been investigated using a feed-forward-feed-back model of the dependent variables, which interact linearly following first order kinetics (13). 

Let us assume that the light frequency uj is in the vicinity of a well-isolated dipole-allowed exciton resonance. In this case the operator of the exciton-phonon interaction, linear with respect to the operator of displacements of molecules from their equilibrium positions, has the form (3.155). It can be shown that in the first order of perturbation theory the real part of 7 uj, k) is given by the following formula (we assume here that the crystal temperature T = 0) ... 

See, e.g. Interactive Linear Algebra A Laboratory Course Using Mathcad, G. J. Porter and D. R. Hill, Springer Verlag, New York, 1996. 

This is true for dilute products with non-interacting linear isotherms. It is more accurate, especially for more concentrated products, to balance the solvent fractions in the feed and in the eluent streams. 

The retention and the selectivity of separation in RP and NP chromatography depend primarily on the chemistry of the stationary phase and the mobile phase, which control the polarity of the separation systems. There is no generally accepted definition of polarity, but it is agreed that it includes various selective contributions of dipole-dipole, proton-donor, proton-acceptor, tt-tt electron, or electrostatic interactions. Linear Free-Energy Relationships (LFER) widely used to charactaize chemical and biochemical processes were successfiiUy apphed in liquid chromatography to describe quantitative structure-retention relationships (QSRR) and to characterize the stmctural contributions to the retention and selectivity, using multiple linear correlation, such as Eq. 

For the purpose of understanding the characteristics of the atmosphere s behavior at certain scales, it is often useful to isolate a particular scale and neglect the nonlinear interactions. Linearization enables the study of specific types of fiuid perturbations, their propagation and dispersion characteristics, and the scale at which fiuid instabilities are likely to be observed. 

Hamiltonian of orhit-lattice interaction, linear in lattice variables. The response of a paramagnetic crystal to different external perturbations (electric or magnetic field, hydrostatic pressure, uniaxial pressure), the dependence of spectra on temperature, and the spin-phonon interaction, are all determined by diffo-ent combinations of these parameters. 

The frequency of donor...Cg distance (D) distributions show typical strong bond characteristics for C-H whereas the H...Cg distance (d) distributions and the normaUzed angle distributions reveal the weaker nature of the interaction. Linear nature, as is true for any hydrogen bond, is observed in the angular distribution of O - H and N - H donor species. Comparatively the indole acceptor C-H...jt is more linear than imidazole or isoxazole acceptors, this may be attributed to the increased arene size of the acceptor. The stability ranking as deduced from all the frequency distributions is given as X- H...indole > X- H...imidazole > X- H...isoxazole. 

---