Hopping index

The exponent n, which is known as the hopping index, is actually equal to / d + 1) where d is the dimensionality. Hence, for two-dimensional systems n = /3. Strictly speaking, Eq. 7.12 holds only when the material is near the M-NM transition and at sufficiently low temperatures. At high temperatures, conduction proceeds by thermal excitation of electrons or donors into the conduction band, or injection of holes or acceptors into the valence band. 

A more recent review of the properties of this material has been given by von Molnar and Penney (1985 see also von Molnar et al 1983, 1985). Results discussed in this article, involving the effects of disorder and electron-electron interaction, are described in Chapters 5 and 9. Briefly, the semiconductor-to-metal transition in an increasing magnetic field leads to a conductivity, at 300 mK, that increases linearly with H (von Molnar et al 1983). This is shown in Fig. 3.7. Hopping conduction is observed with an index indicating the influence of a Coulomb gap (Washburn et al 1984), and near the transition a temperature dependence of a as a+mT, with m positive (von Molnar et al 1985). 

Other experimental evidence that the index for is unity is scanty. If we are right in thinking that hopping conduction near the transition is not affected by a Coulomb gap (Chapter 1, Section 15), evidence can be obtained from this phenomenon Castner s group (Shafarman et ai 1986) give evidence for v = 1 in Si P. For early work in a two-dimensional system see Pollitt (1976) and Mott and Davis (1979, p. 138), which give evidence that v = 1. In two-dimensional systems there is evidence (Timp et ai 1986) that the Coulomb gap has little effect on hopping conduction. 

Suppression rules. Let X(p,Qk) denote the short-time Fourier transform of x[ri, where p is the time index, and Qk the normalized frequency index (0t lies between 0 and 1 and takes N discrete values for k = 1,N, Wbeing the number of sub-bands). Note that the time index p usually refers to a sampling rate lower than the initial signal sampling rate (for the STFT, the down-sampling factor is equal to hop-size between to consecutive short-time frames) [Crochiere and Rabiner, 1983]. 

Here indexes a = 1,2 numerate spinor components. Because we ignore direct hopping of d-electrons on the lattice, in the Hamiltonian (10) a quadratic term with the A -operators is absent, and A -operators enter linearly via the hybridization term. However in the second order perturbation with respect to hybridization such a term should appear it describes the induced hopping on the lattice. 

Structure-activity similarity (SAS) maps, first described by Shanmugasundaram and Maggiora (35), are pairwise plots of the structure similarity against the activity similarity. The resultant plot can be divided into four quadrants, allowing one to identify molecules characteristic of one of four possible behaviors smooth regions of the SAR space (rough), activity cliffs, nondescript (i.e., low structural similarity and low activity similarity), and scaffold hops (low structural similarity but high activity similarity). Recently, SAS maps have been extended to take into account multiple descriptor representations (two and three dimensions) (36, 37). In addition to SAS maps, other pairwise metrics to characterize and visualize SAR landscapes have been developed such as the structure-activity landscape index (SALI) (38) and the structure-activity index (SARI) (39). 

Let us now refer back to Fig. 10(b) to set up a formal indexing system for the system of potential energy maxima and potential energy minima. Then the equations for transport across the series of potential energy barriers can be identified for individual barriers, and the relationship between the equations can be examined. It is clear that the area densities effective for forward hopping over one barrier will be the same area... 

In the sums above, each sum is prime with respect to (i.e. excludes) the index of the previous sum, except for the inner-most sum, which is also prime with respect to i. For example, in the 4-hop term, k can equal i, but not j. Similarly, 1 can equal j, but not k or i. 

The Lattice Boltzmann Method (LBM), its simple form consist of discreet net (lattice), each place (node) is represented by unique distribution equation, which is defined by particle s velocity and is limited a discrete group of allowed velocities. During each discrete time step of the simulation, particles move, or hop, to the nearest lattice site along their direction of motion, where they "collide" with other particles that arrive at the same site. The outcome of the collision is determined by solving the kinetic (Boltzmann) equation for the new particle-distribution function at that site and the particle distribution function is updated (Chen Doolen, 1998 Wilke, 2003). Specifically, particle distribution function in each site f[(x,t), it is defined like a probability of find a particle with direction velocity. Each value of the index i specifies one of the allowed directions of motion (Chen et al., 1994 ThAurey, 2003). 

However, in order to satisfy the requirements of Hop and Hos/ the parameters i s and stand on the following conditions (1) Us=[m+(l/2)]. and Vp=mn iot s=100% and 7p=0) (2) Ds=m/rand t =[m-(l/2)] r(for r]s=0 and =100%), where m is a positive integer. Under these conditions, the values of related parameters m, ffrT, and Ni are listed in Table 1. In order to fulfill the required TIR inside the substrates, only conditions at m=l are valid. However, the feasibility of fabricating these elements is usually limited by the finite refractive index modulation strength n of a recording material. Therefore, an alternative design method is described below to overcome this drawback. 

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