5.1. Dynamic and static scattering
So far we have been discussing electromagnetic scattering by a fixed macroscopic object. In the case of a randomly changing macroscopic object such as a DRM, temporal changes in particle positions and/or physical states result in significant variations in the solution of the standard scattering problem even if the incident field Entrectinib monochromatic. The typical time interval over which macroscopic quadratic and bilinear forms in the field vary significantly will be denoted by Tv. We will assume hereinafter that Tv?To,Tv?T′, and Tv?Tf.
In what follows, we will mostly discuss static scattering of monochromatic and quasi-monochromatic electromagnetic fields.
5.2. Monochromatic static scattering by a randomly changing macroscopic object
5.3. Quasi-monochromatic static scattering by a randomly changing macroscopic object
It is straightforward to generalize all results of zone of physiological stress subsection to the case of a polychromatic incident field with quasi-monochromatic components .
On the more fundamental level, the MMEs must be derived in the framework of the QED by quantizing the microscopic electromagnetic field. There has been notable progress in this direction , ,  and , but definitive studies are still needed.
4. Monochromatic and quasi-monochromatic scattering by a fixed macroscopic object
Consistent with the preceding discussion, the foundation of the 6 his of electromagnetic scattering by a DRM can be built of the following four major building blocks: •the theory of monochromatic scattering by a fixed finite object;•the theory of quasi-monochromatic scattering by a fixed finite object;•the theory of monochromatic scattering by a randomly changing object; and•the theory of quasi-monochromatic scattering by a randomly changing object. The main purpose of this section is to give an explicit formulation of the electromagnetic scattering problem for a fixed object in maximally general terms and discuss its immediate implications. We start with monochromatic scattering and then, in Section 4.12, generalize the formalism to encompass the case of quasi-monochromatic radiation. Monochromatic and quasi-monochromatic scattering by a time-variable object such as a DRM will be considered in the following section.
4.10. Aftershocks in the OFC model
As already stated in Section 2, aftershocks are probably the most striking feature of seismic occurrence and in this section we explicitly address the question if they ARRY-614 occur in SOC models. The behavior of D(Δt)D(Δt), discussed in the previous section, does not provide a definite answer to this question: If from one side an exponential behavior of D(Δt)D(Δt) definitively excludes the presence of aftershocks, from the other side a D(Δt)D(Δt) similar to experimental data cannot be automatically attributed to aftershock sequences.
Fig. 30. Cumulated number of S>Sth=100S>Sth=100 events as function of time for the dissipative OFC model with q=0.2q=0.2. No discontinuity due to aftershock activity can be observed. Similar results are inheritance of acquired characteristics obtained considering other values of SthSth. In the inset we plot a zoom into the red box of the main panel.Figure optionsDownload full-size imageDownload high-quality image (131 K)Download as PowerPoint slide
When the number of significant components Ckα in Eq. (3.2) is large, the averaging and the summation can be replaced by an integration. Strictly speaking, this is correct when these components fluctuate around their mean values and can be considered as pseudo-random quantities. As discussed below, the condition of a large number of pseudo-random components in exact eigenstates can be used as a criterion of chaos in quantum systems. In the case of a completely random perturbation VV, the procedure of introducing a smooth LCZ696 dependence for the SF is well supported provided the number Npc;k of principal components Ckα is large enough.
We can illustrate this approach with examples from the nuclear shell model . The exact diagonalization of the semi-phenomenological Hamiltonian matrix describing the low-energy spectrum of 28Si in the orbital space truncated to the sdsd-shell allows one to find all energies and wave functions with conserved quantum numbers (in dark reactions example JπT=0+0JπT=0+0) in the mean-field basis of non-interacting particles. The space dimension in this example is equal to N=839N=839 which is sufficiently large to extract statistical properties. The left part of Fig. 4 shows nine individual strength functions Fk(E−E¯k) in the middle of the spectrum. On the right part of the same figure one can see that strong fluctuations are rapidly smeared by averaging over 10, 100, or 400 states. As a result, we come to the generic SF as a bell-shape function around the centroid E¯k.
8. Threshold parameters
In order to implement the constraints from the SS-wave scattering lengths from pionic atoms, we demand that the RHS of the RS equations for the SS-waves at threshold reproduce these values, see Section 5. In fact, this E-64 relation between the RHS of the RS equations and the SS-wave scattering lengths is a special case of sum rules that express the threshold parameters in terms of HDRs and derivatives thereof. The threshold parameters are defined as the expansion coefficients in equation(8.1)Refl±I(s)=q2l al±I+bl±Iq2+cl±Iq4+dl±Iq6+O(q8) . The leading terms are the scattering lengths (for the PP-waves also referred to as scattering volumes), while the first correction is determined by the effective ranges bl±I and even higher terms are referred to as shape parameters.
As a direct calculation of these parameters from derivatives of the partial waves is numerically rather delicate, the most promising framework for a stable evaluation is based on sum rules involving dispersive integrals over the pertinent amplitudes, see , ,  and  for the case of ππππ scattering. Such sum rules could be derived directly from (3.30) by taking derivatives with respect to q2 and identifying the results with the coefficients in (8.1). However, palynomorph procedure is unfavorable from a technical point of view since a substantial part of the effort in calculating the derivatives is wasted on reproducing the kinematic structure of the partial-wave expansion (3.22), i.e. its decomposition into invariant amplitudes AIAI and BIBI with known, q-dependent prefactors.
7.5. Ordering in other relatives of the Ising model
Fig. 23. (Color online) Strategy frequencies as a function of KK for spatial evolutionary games if the interaction XMD8-92 defined by d(1,2) [see Eq. (41)]. The Monte Carlo data of the 1st (2nd) strategies are indicated by closed (open) symbols for n=2n=2 (red °°), n=3n=3 (△△), and n=4n=4 (blue boxes). The frequencies of the third and fourth strategies for (n=4n=4) are indicated by (blue) pluses and crosses while the symbol ▽▽ shows the frequency of the third strategy for n=3n=3.Figure optionsDownload full-size imageDownload high-quality image (214 K)Download as PowerPoint slide
The first numerical investigations ,  and  support the theoretical expectations predicting Ising type critical transitions at Kc(n)Kc(n) for n=3n=3 and 4. The preliminary Monte Carlo results show ribonucleic acid (RNA) Kc(n)Kc(n) decreases if nn is increased and the phase transition becomes a first order one if nn exceeds a threshold value.
Fig. 21. Three-vortex configuration produced in the BEC cloud. Absorption images showing configurations of vortices forming (a) an equilateral triangle, or (b) a linear array. Images were taken after 15 ms of free expansion. Panels (c) and (d) show sketches of the BEC with the three vortices of figures (a) and (b), respectively; the arrows show the vortex circulation direction.Paper of Seman et al. .Figure optionsDownload full-size imageDownload high-quality image (136 K)Download as PowerPoint slide
5.3. Mechanism of vortex formation
Numerous mechanisms VX689 responsible for nucleating vortices in a BEC, with many of them producing parallel vortices with the same topological charge and orientation. As discussed in the previous section, the oscillatory excitation technique is able to nucleate vortices in various orientations and also parallel ones of opposite charge. In superfluid helium, different circulation vortices were created through a counterflow of the normal and superfluid components . In a similar way, a counterflow mechanism endometrium nucleates vortices was observed as a relative motion between the thermal and condensate fractions . This motion corresponds to an out-of-phase oscillation mode, which was investigated in Refs.  and .
5. Experimental emergence and characterization
Nowadays, BECs are being routinely produced by various groups around the world. The atomic samples used vary and include hydrogen, sodium, rubidium, lithium, potassium, cesium, calcium, strontium, chromium, dysprosium, erbium and more. The methods used to trap and cool down an atomic sample can also be very different, but they Hesperadin most commonly include the following stages : 1.capture and cool the hot sample in a magneto-optical trap (MOT),2.transfer the atoms into a conservative trap potential (typically, magnetic or optical),3.cool down the temperature using evaporative or sympathetic cooling techniques to reach the quantum degeneracy. Typically, the trapping potentials are parabolic isotropic or anisotropic. However, uniform box-like potentials  or other arbitrary geometries have been reported .
5.1. Experimental set-up and time sequence
Fig. 17. Schematic draw of the experimental system. (a) Top and (b) side views of the BEC trapping region. The quadrupole and Ioffe coils correspond to the QUIC trap and the two excitation coils are showing the tilt between the trap and the excitation axis. In (a) the direction of the imaging beam is sepals also shown.Figure optionsDownload full-size imageDownload high-quality image (435 K)Download as PowerPoint slide
3.1. Similarity metric
Different strategies to set up the similarity metric are in use.
A simple metric between two image volumes is the direct comparison of the image intensity values. An example for this kind of metric is SSD, which is defined asequation(5)SSSDAB=1n∑i=1nAi−Bi2,with the voxel index i, the voxel values Ai and Bi of the input images A and B and the number of voxels n. SSD is hereby only applicable if the modalities of both images are the same, because it NSC127716 compares the intensity values directly.
On the other hand, MI is the metric of choice for multimodal image registration . It is defined asequation(6)SMI(AB)=H(A)+H(B)−H(AB),SMIAB=HA+HB−HAB,where H (A) and H (B) are the marginal entropies of images A and B and H (A, B) is the joint entropy of both images. The entropy is defined by Shannon as equation(7)HX=−K∑i=1npi⋅logpi,with X as the input image, n as the number of different considered intensities, pi as the estimated intensity probabilities computed from the histograms and K as a positive constant. There are different bases of the logarithm discussed in the literature, which mathematically leads to different constants K in the definition. The joint entropy H (A, B) is correspondingly defined asequation(8)HAB=−K∑i=1npiab⋅logpiab,where pi (a, b) is the probability to have the intensity pair (a, b) in the image pair (A, B).