Numerical results Clause Samples
Numerical results. In this section we first consider a stock loan contract with an automatic ter- mination clause a (a ∈ [0, q]), r = 0.05, γ = 0.07, σ = 0.15, δ = 0.01, q = 100 and S 0 = 100. We will give six numerical examples to show that how the liquidity, optimal strategy b(a), initial value fa(x) and initial cash q − c depend on automatic termination clause a, respectively.
Example 7.1. We see from graph1 below that the liquidity obtained with automatic termination clause is larger than the circumstance without the automatic termination clause. When the initial stock price S 0 = 100 and a = 100, the client just sell the stock to the bank by the stock loan contract with automatic termination clause. Figure 1. γ = 0.07, r = 0.05, σ = 0.15, δ = 0.01, q = 100, S 0 = 100
Example 7.2. We see from graph 5 below that b is an function of a. Both the client and the bank will take the deal when the initial stock price is in
Numerical results. The numerical technique we use is the same as has been used before for the spinless kicked rotator [50, 51]. A combination of an iterative procedure for matrix inversion and the fast-Fourier-transform algorithm allows for an efficient calculation of the scattering matrix from ▇▇▇ ▇▇▇▇▇▇▇ matrix. The average conductance ⟨G⟩ was calculated with the Landauer for- mula (2.15) by averaging over 60 different uniformly distributed quasi-en- ergies and 40 randomly chosen lead positions. The quantum correction
Numerical results. To investigate the validity of the formulations presented in the previous sections for stress analysis of rotors, several illustrative examples including rotating disks with constant and variable thickness; and a complex rotor are analyzed in this section.
3.5.1 Rotating disk with constant thickness
(a) 1B3 (b) 5×20 L9
Numerical results illustrates the system's outage probability against the transmit power for all possible scenarios in a network setting with (a) = 2 and (b) = 3 relays. Our results are numerically optimized with respect to the power splitting parameters ij. We observe that our proposed protocol outperforms the conventional multi-hop model, where each node sends a signal only to its subsequent node through orthogonal channels. Further- more, the figure shows that an increase in the number of hops results in an improvement of the outage probability performance, with the cases of = 1 and = 2 revealing the most significant difference. In addition, it can be seen that as the number of hops increases the diversity gain is also improved, which complies with our analysis indicating a diver- sity order equal to . Finally, we can see that the theoretical values perfectly match to the simulation results and this observation validates the accuracy of our analysis.
Numerical results. In this section we demonstrate that using HyBR to solve the regularized least squares problem at each ▇▇▇▇▇-▇▇▇▇▇▇ iteration of the variable projection method can be beneficial to super-resolution and blind deconvolution imag- ing applications. More specifically, we show that one can achieve sufficient objective function and gradient norm decrease, as well as more accurate pa- rameter estimation, by using the HyBR method in a reduced ▇▇▇▇▇-▇▇▇▇▇▇ framework. Furthermore, we illustrate that sufficient reconstructions can be computed without requiring a priori selection of a regularization parameter.
Numerical results. To investigate the validity of the formulations presented in the previous sections for stress analysis of rotors, several illustrative examples including rotating disks with constant and variable thickness; and a complex rotor are analyzed in this section.
3.5.1 Rotating disk with constant thickness
(a) 1B3 (b) 5×20 L9
3.6 Discretizing a constant thickness disk into one B3 finite element along the axis with a distribution of L9 elements over the cross-section. In this example, the effect of the various discretizations associated with the CUF approach on stress and displacement fields of the disk was investigated. Considering 1B3 along the axis, radial displacement (ur), radial stress (σrr) and circumferential stress (σθθ) for the point located at the mid-radius of the disk (r = 0.1524 (m)) for different distributions of LEs on the cross-section are presented and compared with analytical and converged 3D conventional finite element (FE) solutions in Table 3.1. The closed form formulation of the analytical solutions, which are reported in the second row of Tables 3.1 and 3.2, for annular disks with constant thickness can be found in Ref. [62]. The 3D conventional FE solution, in the last row of these tables, was performed by the ANSYS FE package. Several solid models with finer meshes especially at vicinity of the inner and outer radii of the disk were analyzed in order to check convergence of the ANSYS model. In the converged ANSYS model, the geometry of the disk has been meshed into 1920 SOLID185 (3D 8-node structural solid) elements as 24×40 elements on the cross-section (24 elements in radial direction and 40 element in circumferential direction) and 2 elements across the thickness of the disk, and in this case, the total number of DOFs is equal to 9000. In this dissertation, the computational costs of FE models are provided in terms of DOFs.
Table 3.1 Displacement and stresses at mid-radius of the rotating disk with constant thick- ness for different LEs on the cross-section. ur (μm) σrr (MPa) σθθ (MPa) 85.93 276.70 201.42 81.56(5.07) 265.45(4.06) 186.07(7.62) 81.39(5.28) 270.27(2.32) 185.54(7.88) 81.50(5.15) 270.70(2.16) 185.73(7.79) 81.98(4.59) 272.08(1.66) 189.37(5.98) 83.53(2.79) 273.72(1.07) 194.00(3.67) 84.91(1.18) 275.33(0.49) 195.93(2.72) 85.19(0.86) 276.45(0.08) 198.05(1.67) 85.19(0.86) 276.58(0.04) 197.88(1.75) 85.23(0.82) 275.88(0.29) 199.01(1.19) 85.29(0.74) 275.55(0.41) 200.10(0.65) 4×32 L4 1440 4×40 L4 1800 4×48 L4 2160 5×32 L4 1728 1D CUF...
Numerical results. In this section we illustrate the effectiveness of using the W-GCV method in Lanczos-hybrid methods with Tikhonov regularization.
2.5.1 Results on Various Test Problems
Table 2.1: Results of using G^(k) to determine a stopping iteration. The numbers reported in this table are the iteration index at which our Lanczos- hybrid code detected a minimum of G^(k). Comparing the results in Table 2.1 with the convergence history plots shown in Figure 2.7, we see that our approach to choosing a stopping iteration is very effective. For illustration purposes, we provide in Figure 2.8 the true and blurred satellite image, followed by the reconstructed image after 197 iterations of the Lanczos-hybrid method using Tikhonov and W-GCV. Although the scheme does not perform as well on the Baart example, the results are still quite good considering the difficulty of this problem. (Observe that with no regularization, semi-convergence happens very quickly, and we should therefore expect difficulties in stabilizing the iterations.) These results show that our adaptive W-GCV method performs better than standard GCV,
(a) True Image (b) Observed Image
Numerical results. In this part we investigate the PHYLS performance of FD and its HD counterpart in presence of both colluding and non-colluding EDs. The spatial density of the small-cell BSs is set to be λ (d) = 4 per km2. The (per-user) BS and UE transmit powers are kept fixed at pd = 23 dBm and pu = 20 dBm, respectively. The noise spectral density at all receivers is −170 dBm/Hz and the total system bandwidth is W = 10 MHz. The MC simulations are obtained from 20 k trials in a circular region of radius 10 km. The results are taken over two resource blocks. In the FD small-cell network, the DL and UL run simultaneously, whereas in the HD small-cell network, the DL and UL occur over different resource blocks. Furthermore, in the FD system, we take into account different interference cancellation schemes. In particular, in the DL, we consider the cases with and without SIC capability at the UE side. More- over, in the UL, we capture the performance under different perfect SI cancellation and NLOS residual SI with a variance of −55 dB [5].
Numerical results. The calculated response from the acceleration measure- ments in the mine were given as vectors of node displace- ment vs. time. The adhesive stress loads between rock and shotcrete was calculated by use of the stiffness of the elastic springs in the model. The results from eight sets of measurement data are compiled in Figures 8–9. The adhesive stresses between shotcrete and rock, perpendicular to and parallel with the rock surface, are shown as one bar representing each measurement point, i.e. x = 4.5, 7.5, . . . ,