Euler modified

Various integration methods were tested on the dynamic model equations. They included an implicit iterative multistep method, an implicit Euler/modified Euler method, an implicit midpoint averaging method, and a modified divided difference form of the variable-order/variable-step Adams PECE formulas with local extrapolation. However, the best integrator for our system of equations turned out to be the variable-step fifth-order Runge-Kutta-Fehlberg method. This explicit method was used for all of the calculations presented here. 

EulerS modified Arrhenius s formula for a mixture, tj A B C. . ., assuming incomplete ionisation, and represented the viscosity of a salt solution by ... 

The power developed by the flow in a reaetion turbine is also given by the general Euler equation. This equation ean be modified for maximum utilization... 

The result is a modified Euler number. You can prove to yourself that the pressure drop over the particle can be obtained by accounting for the projected area of the particle through particle size, S, in the denominator. Thus, by application of dimensional analysis to the force balance expression, a relationship between the dimensionless complexes of the Euler and Reynolds numbers, we obtain ... 

The modified Euler method needs two initial values y and y, and is given by... 

A combination of open- and closed-type formulas is referred to as the predictor-corrector method. First the open equation (the predictor) is used to estimate a value for y,, this value is then inserted into the right side of the corrector equation (the closed formula) and iterated to improve the accuracy of y. The predictor-corrector sets may be the low-order modified (open) and improved (closed) Euler equations, the Adams open and closed formulas, or the Milne method, which gives the following system... 

In addition, it is dubious whether this new correlation due to Brucato et al. (1998) should be used in any Euler-Lagrangian approach and in LES which take at least part of the effect of the turbulence on the particle motion into account in a different way. So far, the LES due to Derksen (2003, 2006a) did not need a modified particle drag coefficient to attain agreement with experimental data. Anyhow, the need of modifying particle drag coefficient in some way illustrates the shortcomings of the current RANS-based two-fluid approach of two-phase flow in stirred vessels. 

Again, it is always possible to define a gauge where the 00 part of the Einstein equations reduces exactly to the Poisson equation, but at the cost of drastically modifying the mass conservation and the Euler equations. 

Derivation of a ray path for the geometrical optics of an inhomogeneous medium, given v(r) as a function of position, requires a development of mathematics beyond the calculus of Newton and Leibniz. The elapsed time becomes a functional T [x(f)] of the path x(r), which is to be determined so that ST = 0 for variations Sx(t) with fixed end-points Sxp = Sxq = 0. Problems of this kind are considered in the calculus of variations [5, 322], proposed originally by Johann Bernoulli (1696), and extended to a full mathematical theory by Euler (1744). In its simplest form, the concept of the variation Sx(t) reduces to consideration of a modified function xf (t) = x(f) + rw(f) in the limit e — 0. The function w(f) must satisfy conditions of continuity that are compatible with those of x(r). Then Sx(i) = w(l)dc and the variation of the derivative function is Sx (l) = w (f) de. 

A modified effective Hamiltonian Goep is defined by replacing vxc by a model local potential vxc(r). The energy functional is made stationary with respect to variations of occupied orbitals that are determined by modified OEL equations in which Q is replaced by Goep- 84>i is determined by variations 8vxc(r) in these modified OEL equations. To maintain orthonormality, <5, can be constrained to be orthogonal to all occupied orbitals of the OEP trial state , so that (r) = J]a(l — na)i). First-order perturbation theory for the OEP Euler-Lagrange equations implies that... 

We present how to treat the polarization effect on the static and dynamic properties in molten lithium iodide (Lil). Iodide anion has the biggest polarizability among all the halogen anions and lithium cation has the smallest polarizability among all the alkaline metal cations. The mass ratio of I to Li is 18.3 and the ion size ratio is 3.6, so we expect the most drastic characteristic motion of ions is observed. The softness of the iodide ion was examined by modifying the repulsive term in the Born-Mayer-Huggins type potential function in the previous workL In the present work we consider the polarizability of iodide ion with the dipole rod method in which the dipole rod is put at the center of mass and we solve the Euler-Lagrange equation. This method is one type of Car-Parrinello method. 

DR disengagement ratio, d /d (/ - 1)// modified Euler number, XPjpi ... 

For the modified Euler method, we expand the Taylor series as... 

There are two methods of obtaining a curve of r vs x from Eqs. (B) and (D). The first approach is to write Eq. (B) in difference form for a small change in conversion. Ax, and solve by stepwise numerical integration. As an illustration let us follow through three incremental calculations using the modified Euler method. We write Eq. (B) as... 

It is convenient to nondimensionalize. We must identify a characteristic length scale tc and velocity scale uc. For the former we choose d, though we recognize that this would need to be modified if the fluid is unbounded. For the characteristic velocity scale, it is traditional in this field to choose the maximum value of Uf In addition, we assume that the characteristic time is tc = ic/uc, and the characteristic pressure is pc = pu2c. Then the governing equations, which are the continuity equation and the inviscid Navier-Stokes equations (usually called the Euler equations), can be written in the form 3u... 

When the EKR system is enhanced by the addition of an acid, these equations should be adequately modified in the same manner as was done for the onedimensional model. Rnally, the new transient values of the concentrations at the time t + At can be calculated using a numerical solution of differential equations Uke Euler s method ... 

An additional limitation of the index theorem is that in higher dimensions, there may often be different time scales corresponding to fast and slow reactions, so that the dynamics rapidly relax to a manifold that is embedded in the N-sphere. In this case, the Euler-Poincare characteristic of the submanifold may be different from that of the iV-sphere and (14) would have to be modified appropriately. A very general mathematical approach in which the geometric and topological properties chemical kinetic equations is treated has recently been developed. Although the formulation is in principle capable... 

In the presence of rapid molecular motion, the NMR line shape is modified by the reduced EFG tensor depending on the rate of reorientation and the orientations of the principal axes of the EFG tensor relative to the rotation axis. A calculation of the averaged tensor is performed by transformation of the EFG tensor in motion to a final reference frame in terms of Euler angles a, and y [151,154,155]. The spectral frequency in the fast... 

Mazur et al. [103, 104] demonstrated the conformational dynamics of biomacromolecules. However, their method scaled exponentially with size and relied on an expensive expression for the inter-atomic potentials in internal coordinates. Subsequently, our group pioneered the development of internal coordinate constrained MD methods, based on ideas initially developed by the robotics community [102, 105-107], reaching 0(n) serial implementations, using the Newton-Euler Inverse Mass Operator or NEIMO [108-110] and Comodyn [111] based on a variant of the Articulated Body Inertia algorithm [112], as well as a parallel implementation of 0(log n) in 0(n) processors using the Modified Constraint Force Algorithm... 

The connectivity h of a polyhedral fragment is one greater than the number of holes and satisfies the modified Euler s relationship v+t-e=3-h. 

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