Rational frequency ratio

One application of an analog oscilloscope is in examining the relative phase relationship of two sinusoidal wave forms, one of which (of frequency is supplied to the vertical input (/) of the oscilloscope, the other (of frequency /j/) to the horizontal input (x). When the display mode is chosen to be. v vs. /, the. v voltage replaces the internal sawtooth sweep voltage and the oscilloscope becomes a sensitive device for comparing the frequencies of the sine waves. When the ratio of the frequencies is a rational fraction, a symmetrical closed pattern called a Lissajous figure will appear on the screen. The frequency ratio can be obtained from the form of the pattern by using the formula... 

The corresponding trajectories can be best visualized as motion restricted to a two-dimensional torus, as shown in Fig. 1. If the frequency ratio, or the winding number ( i/( 2, is a rational number, the two DOFs are in resonance and an individual trajectory will close on itself on the torus. By contrast, if coi/a)2 is an irrational number, then as time evolves a single trajectory will eventually cover the torus. The motion in the latter case is called conditionally periodic. 

Classifications of the 12 landslide associated controlling parameters used in this study are shown in Table 2. Frequency ratio was used for calculating the rating of the relative importance of each factor class to landslide occurrence (Fig. 3). Equation 1 was used for frequency ration calculation ... 

The following characterizes the frequency behavior (not shown) for the MRW states. Both u and 0J2 vary continuously with the parameter a. There is no entrainment (frequency locking) between the two frequencies of MRW states, that is, there are no parameter intervals (steps) over which the frequency ratio is a constant rational number [13]. This is a general feature of MRW solutions first proved by Rand [38]. 

It is important to point out once again that explanations (rationalizations) of isotope effects which employ arguments invoking hyperconjugation and/or steric effects are completely equivalent to the standard interpretation of KIE s in terms of isotope independent force constant differences, reactant to transition state. In turn, these force constant differences describe isotope dependent vibrational frequencies and frequency differences which are not the same in reactant and transition states. The vibrational frequencies determine the partition functions and partition function ratios in the two states and thus define KIE. The entire process occurs on an isotope independent potential energy surface. This is not to claim that the... 

The analytical computation of critical ionization fields starts with an analysis of the widths of resonances in the classical phase space. Resonances occur whenever the ratio of the external driving frequency u and the unperturbed Kepler frequency Cl of the Rydberg electron is rational, i.e. 

There are certain radii that are resonant the ratio of the frequencies is rational,... 

As Table 4.1 shows, several bimetalhc DENs have been employed as homogeneous catalysts, predominately for hydrogenations. The most detailed studies have been with aUyl alcohol hydrogenation and the partial hydrogenation of 1,3-cyclooctadiene (1,3-COD) test reactions. In these studies, turnover frequencies (TOEs), which can be normalized per mole of nanoparticles, can be compared as a function of the metaUic atomic ratio in the DENs. Comparison with physical mixtures of monometalhc DENs with the same net atomic ratio allows investigators to directly compare both the magnitude and direction of changes to rationally prepared control materials. 

The presence of zero-quantum coherence during the mixing time can substantially distort NOE intensities. This coherence can not be removed with phasecycling or gradient pulses while preserving z-magnetisation. A z-filter scheme was proposed earlier for the elimination of the coherence. It utilises an adiabatic inversion pulse with linear frequency sweep applied simultaneously with a gradient. Cano et alP demonstrated that a better suppression is achieved with a z-filter cascade that combines several filter elements. The attainable suppression ration is then equal to the multiplication of the ratios for each element. 

Planck constant Symbol h. The fundamental constant equi to the ratio of the energy of a quantum of energy to its frequency v = frv. It has the value 6.626 176 x 10 J s. It is named after Max Planck. In quantum-mechanical calculations (especially particle physics) the rationalized Planck constant (or Dirac constant) h = hl2% = 1.054 589 x 10 1 s is frequently used. 

The chemical composition of the framework has a definite impact on the acid strength of zeolites and zeotypes. The total number of Bronsted acid sites present in a zeolite depends on the framework Si/Al ratio or in general on the cations. The variation of stretching frequency of bridging hydroxyls [94] and results from H NMR spectra [95] suggest that the strength of Bronsted sites in HY zeolites increases with decreasing Si/Al ratio. This has been rationalized with the concept of... 

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