Numerical Framework

The mathematical method that will be used to characterize the system response is the stroboscopic map that was used by Kai Tomita (1979) and Kevrekidis et al. (1984,1986). If a point in the phase plane is used as an initial condition for the integration of the ordinary differential equations (odes) (4), a trajectory will be traced out. After integrating for one forcing period (from t = 0 to t = T), the trajectory will arrive at a new point in the phase palne. This new point is defined as the stroboscopic map of the original point and integration for 

The conditions for the existence of a pT-periodic solution to the forced system can be written in terms of a fixed point equation for the pth iterate of the stroboscopic map 

The differential equation for M in (7) is non-antonomous and involves evaluation of the jacobian of the forced-model equations at the current value of the trajectory jc(x0, p, t) for each time step so that it must be integrated simultaneously with the system equations. Upon convergence on a fixed point, the matrix M becomes the monodromy matrix whose eigenvalues are those of the jacobian of the stroboscopic map evaluated at the fixed point and are called the Floquet multipliers of the periodic solution. 

The Floquet multipliers determine the stability and character of the fixed point (or limit cycle) much in the same way as the eigenvalues of the jacobian 

Critical fixed points that are undergoing bifurcation can be found by augmenting the set of fixed-point equations (5) with one of these Floquet multiplier conditions  

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Therefore no longer qualitative statements are cast into a numerical framework but easily measurable values are linked to obtain information on complex stability or bond energies. Thus, by linear regression analysis, two new parameters are obtained which -once they are known for sufficiently many different metal ions, both essential and non-essential ones - in turn can be linked to this biochemical property of essentiality. Electrochemical ligand parameters for different complexes of the same metal ion are correlated... 

Micromechanical models continue to be an area of active research with theories and numerical frameworks being modified to incorporate additional microstructural or physical features like polycrystalline structure, defects, or fatigue [46, 47, 48, 49, 50, 51, 52, 53], Recent advances in micro-electromechanical and macroscopic models include t non-linear behavior have been performed by Landis and coworkers [54, 55, 56], Recently, Huber discussed the utility and limitations of micromechanical models [57], and observes that while these descriptions do not account for grain-to-grain interactions, sharp domain wall interfaces are readily incorporated. 

Introduction into the mathematical, numerical framework typically used for FE-simulations ... 

Predicting the solvent or density dependence of rate constants by equation (A3.6.29) or equation (A3.6.31) requires the same ingredients as the calculation of TST rate constants plus an estimate of and a suitable model for the friction coefficient y and its density dependence. While in the framework of molecular dynamics simulations it may be worthwhile to numerically calculate friction coefficients from the average of the relevant time correlation fiinctions, for practical purposes in the analysis of kinetic data it is much more convenient and instructive to use experimentally detemiined macroscopic solvent parameters. 

Importantly for direct dynamics calculations, analytic gradients for MCSCF methods [124-126] are available in many standard quantum chemistiy packages. This is a big advantage as numerical gradients require many evaluations of the wave function. The evaluation of the non-Hellmann-Feynman forces is the major effort, and requires the solution of what are termed the coupled-perturbed MCSCF (CP-MCSCF) equations. The large memory requirements of these equations can be bypassed if a direct method is used [233]. Modem computer architectures and codes then make the evaluation of first and second derivatives relatively straightforward in this theoretical framework. 

The Langevin model has been employed extensively in the literature for various numerical and physical reasons. For example, the Langevin framework has been used to eliminate explicit representation of water molecules [22], treat droplet surface effects [23, 24], represent hydration shell models in large systems [25, 26, 27], or enhance sampling [28, 29, 30]. See Pastor s comprehensive review [22]. 

The many approaches to the challenging timestep problem in biomolecular dynamics have achieved success with similar final schemes. However, the individual routes taken to produce these methods — via implicit integration, harmonic approximation, other separating frameworks, and/or force splitting into frequency classes — have been quite different. Each path has encountered different problems along the way which only increased our understanding of the numerical, computational, and accuracy issues involved. This contribution reported on our experiences in this quest. LN has its roots in LIN, which... 

The weighted residual method provides a flexible mathematical framework for the construction of a variety of numerical solution schemes for the differential equations arising in engineering problems. In particular, as is shown in the followmg section, its application in conjunction with the finite element discretizations yields powerful solution algorithms for field problems. To outline this technique we consider a steady-state boundary value problem represented by the following mathematical model... 

The thermal conductivity of polymeric fluids is very low and hence the main heat transport mechanism in polymer processing flows is convection (i.e. corresponds to very high Peclet numbers the Peclet number is defined as pcUUk which represents the ratio of convective to conductive energy transport). As emphasized before, numerical simulation of convection-dominated transport phenomena by the standard Galerkin method in a fixed (i.e. Eulerian) framework gives unstable and oscillatory results and cannot be used. 

Kamlet-Taft Linear Solvation Energy Relationships. Most recent works on LSERs are based on a powerfiil predictive model, known as the Kamlet-Taft model (257), which has provided a framework for numerous studies into specific molecular thermodynamic properties of solvent—solute systems. This model is based on an equation having three conceptually expHcit terms (258). 

Glue was also used to join the numerous small plywood and wood details necessary to fabricate the structural frameworks typical of early aircraft. Fig. 1 shows a monoplane wing circa 1930 fabricated primarily from wood that was joined via more than one thousand glued joints [2]. Note the many small stringers and gussets bonded together to form the ribs and spars of the wing. 

An important product of the analysis is the framework of engineering logic generated in constructing the models. The numerical estimates of frequencies need only be sufficiently accurate to --------risk-significant plant features. 

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