In Eq Sample Clauses

In Eq. 1, 2 are the mean scattering matrices of the conisered defect+grains model and grains model, resepectively, and 1, 2 are their covariance matrices ( is the average covariance matrix). Then it can be shown that the achievable probability of detection using the proposed approach has a lower bound that is given by 1 − 0.5 × (, ) for a binary classification problem (i.e. defect versus noise) [10]. Here we have assumed that the occurrence of the defect and non-defect cases are equally probable, and as a result, the expected probability of detection is 0.5 when there is no useful characterisation information available in the worst case scenario. However, in practice, large defects tend to have lower BC values which could result in higher probability of detection than small defects. This also indicates that for a given defect, best detection performance can be achieved by minimising the BC between the defect+grains model of the defect and the grains model. Based on the above consideration, we define the detectability index (d-index) of a crack (with size ) as

In Eq. [2], the space-time evolution of the kinematic source process is given by ui(ξ,τ), cijpq denotes the elasticity tensor, νj the fault normal vector, and the last part describes the ▇▇▇▇▇’▇ function derivative. Inserting into [2] the elasto-dynamic ▇▇▇▇▇’▇ functions and a double- couple source description reveals that: • Near-field waves are composed of both P- and S-wave, depend on the temporal slip- evolution on the fault plane, and decay as 1/r3 with distance r; • Intermediate-field exist for P- and S-waves separately, their amplitude and properties depend on the slip function, and their distance decays goes as 1/r2 • Far-field P- and S-waves depend on the slip-rate function and decay as ▇/▇. ▇▇▇▇, ▇▇▇▇-▇▇▇▇▇ (▇▇), ▇▇▇▇▇▇▇▇▇▇▇▇-▇▇▇▇▇ (▇▇) and far-field (FF) terms represent different properties of the wave-field: the near-source motions are more sensitive to the spatio-temporal details of the rupture process, while far-field terms carry the overall signature of the rupture encoded into the moment-rate function. On the other end, the high-frequency far-field motions often exhibit peak ground acceleration (PGA) within the resonance frequency of buildings, and hence are of great interest for engineering purposes. In this context it is of interest to be able to define the (approximate) region in which NF- radiation needs to be included, and where FF-waves dominate. Ground motion simulations codes have been developed to compute either the NF or FF wave-field, with computational costs increasing rapidly if full-wave field simulations are needed. Fully broadband (i.e., including the complete frequency domain of engineering interest) simulation techniques using a single methodology are not feasible due to computational limitations, and hence hybrid approaches are considered (e.g. ▇▇▇ & ▇▇▇▇▇▇, 2003; ▇▇▇▇▇▇ and ▇▇▇▇▇▇▇, 2004; ▇▇▇ et al., 2010; ▇▇▇▇ et al, 2010). An insightful, yet simplified, study has been carried out by ▇▇▇▇▇▇▇▇ et al. (2000) to address the importance of near-field and far-field radiation in layered media for point-source excitation. Their conclusion states that “the importance of near- and intermediate-field terms for the synthesis of complete waveforms can be significant out to regional distances (330 km) at very long periods” (Ichinose et al., 2000). More specifically, near-field and far-field radiation needs to be treated frequency dependent, in that the far-field terms dominate already at short distances for high frequencies, while for low f...

In Eq. 1 parameter a is the growth rate,  represents a retardation factor, which may be defined as  = a0 /z , where z is the asymptotic value of z(t) and is related to the carrying capacity of the system, y0m is the amplitude of the spurt at t = tm , and G(tm , m , t) is the Gauss cumulative function with average value tm and standard deviation  m , defined by: ( tm )2 G(t , , t) = 1 

In Eq ri is the distance from oxygen i to the carbon atom, and λ is a large parameter set at a value of 250. The Gaussians placed along the three collective variables have a width of 0.02, 0,02, and 0.02 Å in the three directions s1, s2, s3 respectively. The Gaussians have a height of 1 kcal/mol and were placed every 100 steps (25 fs).xiii,xiv Each reaction was then modeled by 10 parallel metadynamics simulations of 62 ps in both the forward and the backward direction. Afterwards, for each system all potential energies were collected and placed on the 2D grid of the selected collective variables (Figure 3). A path was fitted from reactant to product on the obtained potential energy surface, and the location of the barrier (in terms of collective variables was determined).

In Eq. 8.7 still dominates over the potential arising from the dimple structure. A first remark we want to make is the textbook [85] result that the classical turning points of the mode p√rofiles can be found from Eq. 8.7 by solving ρ 2 + l2/ρ2 — 2ω˜ = 0, which results in ρ2 = ω˜ ± ω˜ 2 — l2. These points are indicated in Fig. 8.3 for both LG7,1 and LG12,3, and coincide with the bending points on the flanks of the first and the last lobes in the intensity profiles. The second remark is that the model offers the possibility to investigate the effect of aberrations on the mode profiles. As an example, we study spherical aberration, expanding the mirror height profile Δz beyond the quadratic term. Consequently, a fourth order term α = 1/(8kR) L/R (see Chapter 4). On second thought, not only the mirror height profile

In Eq. (1) (|) describes the probability of measuring the noisy scattering matrix from a defect with parameter , the normalisation constant is given by = (∫ (|))−1, and we have assumed that (1) more precisely (i.e. different normalisation constants can be derived for different defects) [3]. The statistical distribution (|) is termed the defect+grains model in this report, and based on Eq. (1) the defect characterisation problem can be formulated by constructing a defect+grains model for each target defect and calculating the conditional probability of them given the measurement. Considering the high dimensionality of a scattering matrix, is converted into the defect principal component (PC) space by the use of principal component analysis (PCA) [5] to obtain pc. The role of PCA is to represent the original dataset in a different coordinate system so that the variance of the data can be more efficiently described using a small number of the coordinate axes. This helps to reduce the dimensionalty of the scattering matrix and as a result, actual defect+grains modelling is performed for pc 1). The PCA process effectively constructs a defect manifold [3] for modelled types of defects, and a large number of noise-free scattering matrices are needed for accurate reconstruction of the defect manifold (i.e. smooth surface for defects defined on 2D parameter spaces). In this report, we consider characterisation of small cracks and holes of sizes 1mm, 2mm, and 3mm, in the frequency range between 1 MHz and 3 MHz. The noise-free dataset used for PCA is obtained by sampling in both the defect size and aspect ratio (defined as the ratio between the width and length of a defect [6]) axes of the parameter space. Defect size is sampled in 0.05λ intervals between 0.05λ and 2λ, and aspect ratio is sampled in 0.1 intervals between 0 (cracks) and 1 (holes). The considered defect sizes fall within the modelled size range of [0.05λ, 2λ] when the frequency is between 1 MHz and 3 MHz. Although defect+grains modelling can be performed for each point on the defect manifold in principle, this is done only for the six cracks and holes in the current work because of the considerable computational time needed for forward modelling. The variation explained by first 10 PC directions is given in Table 1, which indicates that we can use a small number of PCs to represent a defect scatteing matrix without much information loss. 1 48.07 2 11.94 3 1.23 4 0.37 5 0.14 6 0.07 7 0.06 8 0.03 9 0.02 which i...

In Eq. To validate the ef- fectiveness of Multi-graph regularization and iterative views

In Eq the chemical shift scales with B0 according to Eq. [2.3], while the dipolar couplings are field-independent. In particular, if the spinning rate ωr is not much larger than the strength of the anisotropic interactions, residual broadening effects occur. Thus, higher magnetic fields are generally desirable as the dispersion of the signals increases relative to the broadening effects of the dipolar couplings and the resolution is enhanced. In addition, high fields increase the sensitivity of the NMR experiments. Since the chemical shift anisotropy also scales with B0, fast MAS is necessary in high fields. With developments in instrumentation, MAS rates of ~15 kHz are now common practice and higher rates up to ~50 kHz are possible, although at the expense of smaller sample volumes [7, 8]. This is sufficient to reduce the chemical shift anisotropy as well as the homonuclear dipolar couplings between isotope labels such as 13C. Many of the standard cross polarization (CP) experiments can be applied or have been adopted for use with rapid MAS in high field. The CP technique exploits the high abundance, high sensitivity and short relaxation times of the protons by transferring transverse 1H magnetization to another spin species [9]. The maximum enhancement for a 13C signal compared to direct 13C excitation is g1H/g13C ≈ 4. In addition, the recycle delay required for the accumulation of the free induction decays is usually short. Overall, CP introduces a significant gain in sensitivity. During the detection of the signal, heteronuclear decoupling is applied to achieve a high resolution. The robust TPPM sequence is now widely used for this purpose [10]. It uses 180° pulses with alternating phases for efficient decoupling. The CP/MAS experiment with TPPM decoupling is the building block for more advanced techniques, such as two-dimensional correlation spectroscopy.