Secular dynamics

Applying the theorem of Kolmogorov to a real system is not an easy matter. Due to the very strong requests on the smallness of the parameter e (that we have not reported for simplicity in the statement of theorem 4), even using the best available analytical estimates it is typical to end up with ridiculous results. Thus, we resort to the computer-assisted methods of proof. To this end, we consider the so called secular dynamics for the motion of Jupiter and Saturn this will be our model. We proceed in four steps, with the aid of an algebraic manipulator. 

The study of the secular dynamics is the study of the secular part of the Hamiltonian, obtained after an averaging over the mean longitudes. We will restrict ourselves in this text to the case of only two planets. To first-order in the masses, the averaged Hamiltonian is the mean value of H ... 

Michtchenko, T. and Malhotra, R. (2004), secular dynamics of the three-Body problem Application to the v Andromedae planetary system. Icarus (in press). 

Adaughterx/weff-daughter and the daughter will be effectively supported by a greater amount of parent than that in secular equilibrium. These dynamical effects will result in greater U-series fractionation than expected in static systems. 

The sum is called the partially Fourier transformed dynamical matrix, which depends only on Q, and For each wave vector Q the normal mode frequencies of the crystal can be found by setting the secular determinant equal to zero ... 

The system comprised of equations 3.62 and 3.63 has solutions only when the determinant of coefficients and 2 (secular determinant) is zero. The coefficient matrix, or dynamic matrix, is... 

Dynamic melting with initial secular equilibrium... 

If we assume that we do not have any problem with non-dynamical correlation, we may assume that there is little need to reoptimize the MOs even if we do not plan to carry out the expansion in Eq. (7.10) to its full CI limit. In that case, the problem is reduced to determining the expansion coefficients for each excited CSF that is included. The energies E of N different CI wave functions (i.e., corresponding to different variationally determined sets of coefficients) can be determined from the N roots of the CI secular equation... 

Similar methods have been used to integrate thermodynamic properties of harmonic lattice vibrations over the spectral density of lattice vibration frequencies.21,34 Very accurate error bounds are obtained for properties like the heat capacity,34 using just the moments of the lattice vibrational frequency spectrum.35 These moments are known35 in terms of the force constants and masses and lattice type, so that one need not actually solve the lattice equations of motion to obtain thermodynamic properties of the lattice. In this way, one can avoid the usual stochastic method36 in lattice dynamics, which solves a random sample of the (factored) secular determinants for the lattice vibration frequencies. Figure 3 gives a typical set of error bounds to the heat capacity of a lattice, derived from moments of the spectrum of lattice vibrations.34 Useful error bounds are obtained... 

Thus we leam three things 1) the non-crossing rule is not obeyed in the present picture of unstable resonance states, 2) complex resonances may appear on the real axis and 3) unphysical states may appear as solutions to the secular equation. Thus avoided crossings in standard molecular dynamics are accompanied by branch points in the complex plane corresponding to Jordan blocks in the classical canonical form of the associated matrix representation of the actual operator. 

Can we identify the specific features involved in spatiotemporal variability of sources and sinks of carbon over North America on timescales from seasonal to secular, and what are the processes that affect carbon cycle dynamics the most ... 

To account for the radiative decay of CC excited states we consider the density operator p, Eq. (35), reduced to the CC solvent states. It is a standard task of dissipative quantum dynamics to derive an equation of motion for p with a second order account for the CC-photon coupling, Eq. (24) (see, for example, [40]). Focusing on the excited CC-state contribution, in the most simple case (Markov and secular approximation) we expect the following equation of motion... 

Court administrators amassed many of these documents in the context of trials for alchemical fraud. Most historians will be quite familiar with these kinds of criminal records. Scholars of early modern Europe in particular have used them frequently to access the lives of individuals who otherwise did not leave a trace in archives and libraries.14 Historians of science, however, have used trial records much less frequently, particularly those from secular courts.15 These criminal records—particularly the interrogation records—are, of course, notoriously difficult to work with. The voices of accused criminals have come to us in highly mediated form, and the power dynamic involved in confessions recorded under the threat or use of torture makes them particularly problematic. As Natalie Zemon Davis has noted, however, the tales people told, even under the pressure of life and death, "can still be analyzed in terms of the life and values of the person saving his neck by a story."16 Criminal records may never tell us what "really" happened, but the fact that witnesses stories had to... 

Note that the secular equation above can also be written in the dynamic representation... 

A particular effort has been addressed to the study of the dynamics within the dimer and to the characterization of fhe low lying rovibrational states in view of pofenfial inferesf for fhe analysis of spectral features in atmospheric research. Calculations of fhe bound rovibrational states of the dimers have been performed for rofafional sfafes having fofal angular momentum / < 6 by solving the secular problem over the exact Hamiltonian. We have calculated the rovibrational levels for the potential energy surfaces described above of the dimers N2-N2, N2-O2 and for all fhree surfaces (singlet, triplet and quintet) [4,5] of O2-O2. A summary of resulfs and fheir discussion follows. Full accounf of all available data has been given in [5,6,8-10]. 

Figure 14 illustrates the spectra of a stearic spin probe for two greatly different values of the ratio D /D. For very different diffusion limits in the presence of an orienting potential, this shows how sensitive to changes of the order parameter the dynamics of the spin probe are. The slightly different features of the two series of spectra can be ascribed to the modulation of secular terms of the Hamiltonian of Eq. (3.3). The two series of spectra exhibit virtually the same behavior, except that the senes of curves reproduced in Fig. 14 by the continuous lines seem to be more rigid. 

As for the secular trend, two facts, common to all endemic areas, are crucial in assessing any future dynamics of the disease. These are, an apparent shift of the age distribution of the incidence towards the older age groups, and a much longer natural history of the condition compared to previous data. Consequently, it suggests that the intensity of exposure diminished (if still present at all). 

To calculate numerically the quantum dynamics of the various cations in time-dependent domain, we shall use the multiconfiguration time-dependent Hartree method (MCTDH) [79-82, 113, 114]. This method for propagating multidimensional wave packets is one of the most powerful techniques currently available. For an overview of the capabilities and applications of the MCTDH method we refer to a recent book [114]. Additional insight into the vibronic dynamics can be achieved by performing time-independent calculations. To this end Lanczos algorithm [115,116] is a very suitable algorithm for our purposes because of the structural sparsity of the Hamiltonian secular matrix and the matrix-vector multiplication routine is very efficient to implement [6]. 

In deeper waters the deficiency of from its uranium precursor is dramatic (Table 5.5) because this thorium isotope has a very long half hfe (c. 75 000 y) and thus particle scavenging is much more effective at removal than the ingrorvth toward secular equilibrium with Bacon and Anderson (1982) showed that depth profiles of dissolved and particulate °Th could be used to demonstrate the dynamic relationship of metal exchange between particulate and dissolve forms. They argued that the thorium-uranium isotope pair could be used as a tracer of particle removal rates for those metals that faU in the category of adsorbed in Fig. 1.3. 

As argued earlier in the text, 72 (t) serves as an appropriate spatially independent measure of the impact of the dynamical fluctuations on the kinetics of the reaction. The absence of secular terms in 73(1) [cf. Eq. (36)] enables us to treat it as a purely perturbative correction to Jo(t). In such a case, we can discern the impact of the dynamical fluctuations by analyzing... 

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