Matrix model dependence

Once the matrix of dependencies is specified, the model is evaluated at a given metabolic state, characterized by 18 metabolite concentrations and 4 independent flux values. The numerical values for concentrations and fluxes are adopted from Ref. [113], describing the pathway under conditions of light and... 

Therefore, in the RF model we have added higher order terms to the SO model to make it bounded. The explicit form of the RF model depends on the matrix S which should reflect the anharmonicity of the function. Usually we do not have information to specify S in detail and we simply take it to be the unit matrix multiplied by a scalar S.11... 

In Section I we introduce the gas-polymer-matrix model for gas sorption and transport in polymers (10, LI), which is based on the experimental evidence that even permanent gases interact with the polymeric chains, resulting in changes in the solubility and diffusion coefficients. Just as the dynamic properties of the matrix depend on gas-polymer-matrix composition, the matrix model predicts that the solubility and diffusion coefficients depend on gas concentration in the polymer. We present a mathematical description of the sorption and transport of gases in polymers (10, 11) that is based on the thermodynamic analysis of solubility (12), on the statistical mechanical model of diffusion (13), and on the theory of corresponding states (14). In Section II we use the matrix model to analyze the sorption, permeability and time-lag data for carbon dioxide in polycarbonate, and compare this analysis with the dual-mode model analysis (15). In Section III we comment on the physical implication of the gas-polymer-matrix model. 

Comparing the curves in Fig. 2 shows that representing the permeability versus pressure data by either model provides a satisfactory fit to the data over the pressure range of 1 to 20 atm. However, at pressures less than 1 atm. the two models differ in their prediction regarding the behavior of the permeability-pressure curve [Fig. 2]. While the matrix model predicts a strong apparent pressure dependence of the permeability in this range (solid line), the dual-mode model predicts only a weak dependence (broken line). 

Feshbach resonances is purely model dependent since trapping well may exist on one type of adiabatic potential, say in hyperspherical coordinates, while only a barrier may exist on another type, say in natural collision coordinates. However, this is not correct since there are fundamental differences between QBS and Feshbach states. First, the pole structure of the S-matrix is intrinsically different in the two cases. A Feshbach resonance corresponds to a single isolated pole of the scattering matrix (S-matrix) below the real axis of the complex energy plane, see the discussion below. On the other hand, the barrier resonance corresponds to an infinite sequence of poles extending into the lower half plane. For a parabolic barrier, it is easy to show that the pole positions are given by... 

The "variational type of multi-scale CFD, here, refers to CFD with meso-scale models featuring variational stability conditions. This approach can be exemplified by the coupling of the EMMS model (Li and Kwauk, 1994) and TFM, where the EMMS/matrix model (Wang and Li, 2007) at the subgrid level is applied to calculate a structure-dependent drag force. 

Partial least squares regression (PLS) [WOLD et al., 1984] is a generalized method of least squares regression. This method uses latent variables i, 2,. .., i.e. matrix U, for separately modeling the objects in the matrix of dependent data Y, and t, t2,. .., i.e. matrix T, for separately modeling the objects in the matrix of independent data X. These latent variables U and T are the basis of the regression model. The starting points are the centered matrices X and Y ... 

At co-deposition of dielectric and M/SC the structure of a nanocomposite film depends on a relationship between the rate of formation of M/SC particles and the rate of making of a solid dielectric matrix. Models of such deposition are discussed. The special attention is paid to new low-temperature processes of nucleating and growth of nanoparticles in a solid polymeric matrix containing the inclusions of... 

Another model for the sorption and transport of gases in glassy polymers at super atmospheric pressures is the gas-polymer-matrix model, proposed by Raucher and Sefcik (1983). The premise of this model is that the penetrant molecules exist in the glassy polymer as a single population and that the observed pressure dependence of the mobility is completely due to gas-polymer interactions. In the mathematical representation of this model the following expression for sorption and transport is used ... 

Xia, M., Leppert, D., Hauser, S. L., Sreedharan, S. P., Nelson, P. J., Krensyk, A. M., and Goetzl, E. J. (1996a). Stimulus-specificity of matrix metalloproteinase-dependence of human T cell migration through a model basement membrane. J. Immunol. 156, 176-167. 

All theoretical studies on benzoic acid dimer underlined the need for a multidimensional potential surface. These studies have investigated the temperature dependence of the transfer process They included a density matrix model for hydrogen transfer in the benzoic acid dimer, where bath induced vibrational relaxation and dephasing processes are taken into account [25]. Sakun et al. [26] have calculated the temperature dependence of the spin-lattice relaxation time in powdered benzoic acid dimer and shown that low frequency modes assist the proton transfer. At high temperatures the activation energy was found to be... 

Matrix models have a long history in ecology (Chapter 5). They are easy to use and understand and provide a means to project the population-level consequences of effects at the individual level. They cannot represent spatial effects. Including sto-chasticity and density dependence is possible but makes their use more complicated. 

Matrix models are sets of mostly linear difference equations. Each equation describes the dynamics of 1 class of individuals. Matrix models are based on the fundamental observation that demographic rates, that is, fecundity and mortality, are not constant throughout an organism s life cycle but depend on age, developmental stage, or size. Ecological interactions, natural disturbances, or pesticide applications usually will affect different classes of individuals in a different way, which can have important implications for population dynamics and risk. In the following, I will only consider age-structured models, but the rationale of the other types of matrix models is the same. For an example of this approach applied to pesticide risk assessment, see Stark (Chapter 5). 

It can be shown that the population s long-term growth rate only depends on the transition matrix, not on the initial state of the population. If all elements of the matrix are constant, the growth rate can be calculated as the first eigenvalue of the matrix (i.e., the constant that is obtained by solving Equation 3.1). If restricted to constant demographic rates, matrix models do not require any modeling at all, because... 

The simple matrix model depicted earlier is an example of a deterministic matrix model. Deterministic matrix models have no measure of randomness (stochastic-ity), assume constant demographic parameters, and ignore density dependence. Moreover, estimated growth rate and stable stage or age structure refer to an exponentially increasing (or decreasing) population. 

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