and Eq. (8)). We experiment with different learning rates by grid search and find the best to be 0.001 for the meta-optimizer. Training is performed using the common cross entropy loss function, with a batch size of 128. We shuffle the training data set (consisting of 60.000 examples) and divide in two parts equally: the first is used to train a meta-optimizer using both normal data and adversarial examples and the second is used to test its performance while training with normal data and testing with perturbed data. Each experiment ran for 100 steps and the average results are illustrated in Figs. 1 and 2. All experiments are done using α = 0.5 in Eq. (6) during training and α = 0.0 during the meta-optimizer transfer phase, as first introduced in [8]. In addition to XXXX’s performance compared to the meta-optimizer, we evaluate the performance of the meta-optimizer during training and the performance of training a meta-optimizer using α = 1 and testing with α = 0 (L2L and Transfer-NOT labels in Figs. 1 and 2). Figure 1 illustrates the results from generating adversarial examples using the FGSM method (Eq. (7)). The loss functions start from approximately the same value because the networks are always initialized with the same values. In all cases, the meta-optimizer is able to transfer the information learned during training and has comparable performance to XXXX (in some cases per- forming better). We remind that during testing the optimizer uses normal data, but the plots are generated by feeding adversarial perturbed data to the opti- mizee. This implies that the meta-optimizer proposes update rules which lead to smooth surfaces around the tested inputs. Moreover, it is able to learn a robust regularization term during training and transfer it to new tasks without the need to generate new data. Also, the trained meta-optimizer exhibits more stable behavior. This brings evidence that adversarial training leads to more interpretable gradients [24].
and Eq. (4.2), the energy equation can be expressed in terms of the temperature and displacement fields as ρc(t0 + t2)T¨ + ρcT˙ − 2c˜iT˙,i − (κi jT , j),i +t0T0βi ju¨i, j + T0βi ju˙i, j = R + t0R˙
and Eq. 6.5. The heat current w is defined through, ∫ w = 2 fk {ϵ(k) − µ} υkdk (6.6) It follows that for J and w can be written, J = e2K .E + e K . (−∇T ) (6.7) o T 1 w = eK .E + 1 K . (−∇T ) (6.8)
and Eq. (32) on a log scale for η = 0.1, 0.3 and 0.7. We can see that the rate for the encoded protocol decreases substantially compared to the non-encoded protocol. This is mainly due to difference in the exponent N/2 and 2N on λ2(1 λ2) 10−4 in Table I. Unfortunately, p = λ2(1 λ2) 1 for real λ. Hence, it is unlikely that the encoded protocol can per- form better than the non-encoded protocol if the photon creating device is PDC without further resources such as multiplexing. In Fig. 10 we show the dependence of the creation rate (MHz) on λ. However, note that increas- ing λ comes at the cost of creating multiple pairs in the PDCs, which will cause spurious detection events and a severe degradation of the GHZ states. Our analysis is valid only when λ 1. It is reasonable to be hopeful for devices with photon creation probability higher than PDC to be more widely available in the foreseeable future. At the moment, there are other sources of single and entangled photons. For single photons, there are, e.g., trapped ions [35], cold atoms [36] and colloidal CdSe/ZnS quantum dots [37]. For entangled photons, there are, e.g., atomic ensemble [38] and biexciton-exciton cascade quantum dots [39]. Al- though most of these devices are still in the experimental stage, they are evolving rapidly, hence, are good xxxxx- dates for the desired photon sources.
and Eq. (6)into Eq.(5), The best fixed payment is: 0 η* = v +
and Eq. (2.9) it follows that if we choose the correlation function f (n(x)) to sample the d-th power of the local medium density, e.g. f (x) xd, the local mean free path scales linearly with the expectation value of the Delaunay edge length via a constant c:
and Eq. (A.6) then yields the relationship between VEL and the directivity: Re(ZA) λ 2 Dk = Z0 π |7k|2 . (A.7) When accessing the VEL in simulations and measurements it is usual to include losses inside the antenna structure e.g. due to ohmic resistance. In terms of power, these losses are accounted for by multiplying the directivity with a dimensionless efficiency factor ε. If losses inside the antenna structure are included in the VEL, eq. (A.7) relates to the antenna gain G: Re(ZA) λ 2 Gk = ε Dk = Z0 π |7k|2 . (A.8)
and Eq. (7.4). Table 7.1 shows a measured visibility of V = 98.4% for the best-matched geometry of the fiber-detection scheme. In contrast, free-space detection yields only V = 80.0% under the same conditions (d = 14 mm and wp = 68 µm). We can, however, improve the entan- ± ± glement quality attained with free-space detection to that of the fiber-detection scheme, if we detect behind sufficiently small apertures. For instance, we already measure a visibility of V = 90.0% behind 9 mm apertures, whereas we even obtain a value of V = 97.0% be- hind 4 mm apertures. In Figure 7.2 we show the visibilities V45◦ and V135◦ measured as a function of the aperture diameter d for both detection schemes, using wp = 68 1 µm and wf = 65 5 µm. For free-space detection, we clearly observe the “dramatic” increase in visibility with decreasing aperture sizes mentioned above. For fiber-coupled detection, we measure (much) higher visibilities of at least V = 97.5% for all considered aperture sizes. We ascribe this strong contrast in entanglement quality between the two detection schemes to the removal of spatial labeling by the mode-selective character of the fibers. In fact, the fibers select a pure fundamental transverse mode in both r and k-space, irrespective of the aper- ture size, which explains the constantly high visibilities shown in Fig. 7.2. Instead, apertures perform mode selection only in the transverse r-space, which leads to enhanced polarization distinguishability and thus lower visibilities for larger apertures.
and Eq. (8.18) yields NF = N(L/zR)(2/π), where zR = 1 kpw2 is the Rayleigh range of the pump. As we typically work at L/zR ≈ 2.3, the numbers N and NF should be comparable. ≈ ≈ From our experimental results we can estimate the mode number N and Fresnel number NF in three different ways. First of all, we can use Eq. (8.17) and divide the detection angle θdet by the measured diffraction angle θdiff to find N 3 and N 34 for 4 mm and 14 mm apertures, respectively. Secondly, we can use Eq. 8.18 and compare the measured transverse ≈ ≈ coherence length Δxcoh to the aperture size to obtain Fresnel numbers NF 4 and NF 46 for 4 mm and 14 mm apertures, respectively. The third measure for the transverse mode number × × ≈ can be deduced by comparing the single count rates shown in Figs. 8.3(a) and 8.3(b). As fiber-coupled detection per definition addresses a single transverse mode, division of these mentioned count rates yields a mode numbers of N = 34. A similar exercise for a 4 mm aperture (not shown) yields N = 7 104/2.1 104 3. These numbers compare well with the mode numbers N from the first estimate. All estimates show that our experiment addresses typically 4 or 40 modes for the 4 or 14 mm apertures, respectively.
and Eq. 9. The derivatives of Tisurf and Tosurf are defined separately as functions Tisurf der and Tosurf der, the definitions of which are shown below. The model of the room is similarly represented in our Isabelle/UTP encoding by RoomModel. It assigns new values to RAT out and EWT, with RAT out’s value determined using the Euler method. We similarly define Sensor to model the behaviour of the sensor, LimPID to model the behaviour of the PID controller that limits its output to a prede- fined range, and ControlSplitter to model our controlSplitter compo- nent in the Modelica model. These definitions have been created by carefully examining the OpenModelica definitions of the components. The above functions defined in Isabelle/UTP characterise FMUs that to- gether form the FCU system, composed together as described in Section 3. However, our proofs only involve local invariants, so we do not need to con- sider the composition of the FMUs. The local invariants are sufficient to establish those invariants globally, as we discuss in the next section.
