Modulation theories

The validity condition of sudden modulation theory is determined by the requirement of a quick change in J2 in comparison with its conservation time ... 

T. AIbu and C. NdstSsescu, Relative Finiteness in Module Theory (1984)... 

Van Trees, H., Detection, Estimation and Modulation Theory. Wiley, New York, 1968. 

In this section and in the rest of the article we will use the notations and results of Berthelot, Breen and Messing from their book Theorie de Dieudonne Cnstalline II. In fact we will use the methods developed there to define a Dieudonne module theory and prove some theorems on it. 

Proof. The implication from left to right is trivial. To prove the other implication, by Lemma 8.2, we may assume that k is perfect. Let T = Spec(Wn+i(fc)) <- 5 be the closed subscheme of S constructed in Lemma 8.2. Remark that G ObCs0 <=> Gt0 ObCy0 and similar for Ms(G). Hence, by Proposition 7.2, the result follows from the classical Dieudonne module theory. ... 

Actually, however, at least for an isolated XH Y system, the change in length of the H-bonds occurs rhythmically with the frequency of the v(XH Y) vibration. In effect the vKK vibration is frequency modulated by the v(XB. Y) vibration. From a consideration of this more precise classical picture Batuev [29, 30] has shown that the broad band should actually consist of a series of sub-bands of frequencies rXH (XH Y). This is the frequency modulation theory of the origin of the broad vXH bands. This explanation also implies that the band should become narrow at low temperatures when the amplitude of the H-bond stretching vibration is small. 

Prior to the introduction of the frequency modulation theory, a quantum mechanical energy level scheme had been proposed by Stepanov [31], and subsequently developed by this author and his colleagues [32, 33], which also led to the conclusion that the broad vXH bands should, under certain conditions, be resolvable into a... 

These various consequences parallel closely the analogous ones of the fluctuation and frequency modulation theories. There is, however, one important point of difference between the classical and quantum viewpoints which does not seem to have been emphasized previously, namely that transitions from the lowest level of the ground state can occur to several levels of the upper curve. This means that even at very low temperatures, when all the molecules are initially in this lowest energy level, a band of considerable breadth with frequencies rXH + m>(XH Y) will still persist. The temperature independent residual band width is a direct result of the perturbations of the system (in particular the finite change in the distance rxymin) caused by the absorption of a large quantum of radiation of frequency vXH. The same type of explanation may apply to other vibrational bands which remain of finite width at low temperatures the occurrence of such bands have been the cause of considerable discussion [34]. 

Based on the fluorite-type module theory the thermodynamic properties, hysteresis, and reactions between the homologous series can be elucidated and the structures of homologous series experimentally discovered may be modeled. Using these principles a wide range of non-stoichiometric ternary lanthanide higher oxides from RO2 to R2O3 were founded. 

Of the 15 experimentally known phases of the higher oxides only five of them have been determined by X-ray and neutron diffraction using the Rietveld refinements method. To understand the thermodynamic behavior and phase reactions it is helpful to have a model of the undetermined structures. Using the experimental electron diffraction data it is possible to determine the symmetry of the unit cell and develops a transformation matrix between the fluorite and ten of the intermediate phases as shown in Table 2. The module theory provides a method for modeling the unknown structures of the homologous series of the lanthanide... 

Theories based on the Enskog collision time (84) or other solid-like approaches do not have a strongly temperature-dependent frequency correlation time. But they do have a temperature-dependent factor resulting from the need to create the solvent fluctuations in the first place. Thus, all fast-modulation theories predict that the dephasing rate will go to zero at 0 K. 

XVI. Non-Poisson Renewal Processes A Property Conflicting with Modulation Theories... 

XVI. NON-POISSON RENEWAL PROCESSES A PROPERTY CONFLICTING WITH MODULATION THEORIES... 

We define as modulation theory an approach to a noncanonical distribution based on the modulation of processes that with no modulation would yield canonical distributions. For instance, a double-well potential under the influence of white noise yields a Poisson distribution of the time of sojourn in the two wells [150]. In the case of a symmetric double-well potential we have... 

In conclusion, according to the modulation theory we write the waiting time distribution /(f) under the following form ... 

The authors of Ref. 96 have proved that this approach can be used to derive the same analytical expressions as those derived by Barkai [95] in an earlier publication. According to Ref. 97, these analytical formulas can be proven to be equivalent to Eq. (231). In conclusion, Eq. (283) is a reliable way to express aging. We use this approach to prove that superstatistics yields no aging. In fact, the time evolution of p(y, t) using modulation theory reads... 

Batuev, attempting to marshal every argument in favor of his own frequency modulation theory in preference to the predissociation theory, observes that the predissociation theory ignores the fundamental principles of Marxism-Leninism (155). Neither Marx nor Lenin seems to have published any work on the H bond. 

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