Single root models

Generally, single root models rely on solute transport theory to determine the supply of nutrients to the root. Solutes in soil are assumed to be moved by the additive and simultaneous processes of diffusion and convection. The governing equation in radial coordinates is (Nye and Marriott 1969 Barber 1995)... 

A more precise question (Bethke, 1992) is the subject of this chapter in geochemical modeling is there but a single root to the set of governing equations that honors a given set of input constraints We might call such a property mathematical uniqueness, to differentiate it from the broader aspects of uniqueness. The property of mathematical uniqueness is important because once the software has discovered a root to a problem, the modeler may abandon any search for further solutions. There is no concern that the choice of a starting point for iteration has affected the answer. In the absence of a demonstration of uniqueness, on the other hand, the modeler cannot be completely certain that another solution, perhaps a more realistic or useful one, remains undiscovered. 

Thus, for a single-parameter model such as y,j = p + r,j, the estimated variance-covariance matrix contains no covariance elements the square root of the single variance element corresponds to the standard uncertainty of the single parameter estimate. 

Turning now back to the single-root MR BWCC approach, we derive the basic equations for the effective Hamiltonian and cluster amplitudes in the spin-orbital form without the use of the BCH formula. We limit ourselves to a complete model space which implies that amplitudes corresponding to internal excitations (i.e. excitations within the model space) are equal to zero. In our derivation we shall work with the Hamiltonian in the normal-ordered-product form, i.e. 

In Section 6.2, the standard uncertainty of the parameter estimate b0 was obtained by taking the square root of the product of the purely experimental uncertainty variance estimate, sand the (X X) 1 matrix (see Equation 6.3). A single number was obtained because the single-parameter model being considered (yu = /30 4 r1() produced alxl (.Y A T 1 matrix. 

Segers R. and Leffelaar P. A. (2001a) Modeling methane fluxes in wetlands with gas tranporting plants 1. Single-root scale. J. Geophys. Res. 106, 3511-3528. 

On the single root level, Whiting et al. (2003) used an analytical solution of an approximate steady-state model to simulate Zn uptake by T. caerulescens. 

Rhizosphere modeling remains difficult and complex, as it combines technical know-how from several fields such as plant physiology, soil physics, soil chemistry and mathematics. Mechanistic rhizosphere models do not always operate with adequate precision (Rengel, 1993 Darrah and Roose, 2001). Two main fields of application of mechanistic rhizosphere models are carbon flow in the rhizosphere and nutrient uptake by plants. While carbon flow models study the exudation of carbon compounds into the soil and its consequences on the microbial population, uptake models focus on the transport and uptake of ions by roots. In the following sections, we will concentrate on uptake models on the single root scale. 

Figure 2.6.12 Alternative to single root cause incident model...

Approaches which consider one state at a time are often referred to as one-state or state-selective or single-root . They were first proposed in the late 1970s. A paper published by Harris [113] in 1977, entitled Coupled cluster methods for excited states, first introduced the state-selective approach. Four papers which were published in 1978 and 1979 advancing the state-selective approach parts 6 and 7 of a series of papers entitled Correlation problems in atomic and molecular systems part 6 entitled Coupled cluster approach to open-shell systems by Paldus et al. [114] and part 7 with the title Application of the open-shell coupled cluster approach to simple TT-electron model systems by Saute, Paldus and Cfzek [115], and two papers by Nakatsuji and Hirao on the Cluster expansion of wavefunction, the first paper [116] having the subtitle Symmetry-adapted-cluster expansion, its variational determination, and extension of open-shell theory and the second paper [117] having the subtitle Pseudo-orbital theory based on sac expansion and its application to spin-density of open-shell systems. 

Equation (4.94) shows that 17 transforms the projection of the ground state wave function on to the model space, back into the exact ground state wave function. In contrast to the multi-root wave operator,J7, which was introduced in the previous section, our single-root wave operator, 17, is a state-specific wave operator, (see eq. (4.57)). Q does not transform projections of other exact wave functions CP Z a on to the model space into the wave functions for the corresponding exact states. To avoid any confusion we shall use a tilde to distinguish the single-root wave operator. 

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