Discrete jump

Figure Cl.5.8. Spectral jumping of a single molecule of terrylene in polyethylene at 1.5 K. The upper trace displays fluorescence excitation spectra of tire same single molecule taken over two different 20 s time intervals, showing tire same molecule absorbing at two distinctly different frequencies. The lower panel plots tire peak frequency in tire fluorescence excitation spectmm as a function of time over a 40 min trajectory. The molecule undergoes discrete jumps among four (briefly five) different resonant frequencies during tliis time period. Arrows represent scans during which tire molecule had jumped entirely outside tire 10 GHz scan window. Adapted from...

In the SMB operation, the countercurrent motion of fluid and solid is simulated with a discrete jump of injection and collection points in the same direction of the fluid phase. The SMB system is then a set of identical fixed-bed columns, connected in series. The transient SMB model equations are summarized below, with initial and boundary conditions, and the necessary mass balances at the nodes between each column. 

Mixing models based on the CD model have discrete jumps in the composition vector, and thus cannot be represented by a diffusion process (i.e., in terms of and B ). Instead, they require a generalization of the theory of Markovian random processes that encompasses jump processes136 (Gardiner 1990). The corresponding governing equation... 

Due to its large discrete jumps, rotation around the helix axis is not a continuous Gaussian random process, so its actual mean squared angular displacement cannot be used directly in C (t). However, <<5z(/)2> for the equivalent Gaussian random process that makes the same contribution to the FPA is obtained from the relation... 

In this section, we consider the description of Brownian motion by Markov diffusion processes that are the solutions of corresponding stochastic differential equations (SDEs). This section contains self-contained discussions of each of several possible interpretations of a system of nonlinear SDEs, and the relationships between different interpretations. Because most of the subtleties of this subject are generic to models with coordinate-dependent diffusivities, with or without constraints, this analysis may be more broadly useful as a review of the use of nonlinear SDEs to describe Brownian motion. Because each of the various possible interpretations of an SDE may be defined as the limit of a discrete jump process, this subject also provides a useful starting point for the discussion of numerical simulation algorithms, which are considered in the following section. 

For motion by discrete jumps between planes a0 apart, the velocity is the product of the number of jumps per second, T, and the distance aQ of each jump. Equation 13 may now be written as... 

Planck s explanation of the blackbody radiation curves (1900 [4]) and Einstein s explanation of the facts of the photoelectric effect (1905 [7]) indicated that the flow of energy in physical processes did not take place continuously, as had been believed, but rather jerkily, in discrete jumps, quantum by quantum. The contributions of Planck and Einstein were the signal developments marking the birth of quantum theory and the transition from classical to modem physics. 

Several 2D methods in solids make use of discrete jumps in sample orientation, rather than MAS, whereas others change the rate of spinning between evolution... 

Isotactic polypropylene is semicrystalline. In the crystalline regions the molecules form a 3i helix, that is three monomer units form one turn of a helix. This helix performs rotational Jumps around its own axis. After one jump, each monomer occupies the place of its next neighbour before the jump. Molecular reorientation by discrete jumps... 

One-dimensional quadrupole echo NMR lineshape analysis of powder samples is particularly informative when fast, discrete jumps occur between sites of well-defined geometry as, for example, in a phenyl group undergoing two-site exchange. In this case, the characteristic Pake-pattern is transformed into an axially asymmetric lineshape with an apparent asymmetry parameter r] 9 0 (see Equation (6.2.3)) [1-8]. The asymmetric lineshapes, shown on the left in Fig. 6.2.2, can be derived by considering how the individual components of the principal EFG tensor become averaged by the discrete jumps. Within the molecular frame, and in units of as defined by Equation (6.2.2), the static axially symmetric tensor consists of the components = 1, = — 1/2, and V y = — 112. This traceless tensor satisfies the... 

When atoms are excited by X-ray bombardment, inner shell electrons are ejected. The vacancies thus created are immediately filled by outer shell electrons. These outer shell, energetic electrons move toward the lower energy inner shells by discrete jumps from energy level to energy level. The resulting decrease in their energy appears as X-ray photons with wavelengths that are characteristic for each element. 

If the transport is due to discrete jumps of atoms then the diffusion coefficient D is related to the concentration of the jumping species (x), the jumping frequency (v) and the distance over which the jump occurs (d) ... 

Step 2. Use mass and thermal energy balances for both fluids to calculate the slope of conversion and temperature with respect to trx at conditions in the feed. The actual algorithm averages this slope at several positions within a small interval that represents discrete jumps in trx. 

In general, in the development of a separation of a sample mixture, the sample is initially injected onto the HPLC column in a solvent of low elution strength which is then steadily increased until the compounds of interest elute from the column. The main limitation of this method is that it may take a long time to effect elution and consequently it is often quicker to use this procedure in reverse, with discrete jumps in eluent strength (Parris, 1978). Modem equipment allows gradient development to be carried out unattended. 

For this study, we defined innovation as a discrete jump in theory, method, or product. The concept of discontinuity is the important essential in the definition. It should be noted that the definition only refers to measurable outcomes, e.g., theory, method, or product. The definition does not deal with ideas or abstractions that do not enter fields of knowledge in a testable manner. 

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