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# Walker-like domain wall breakdown in layered antiferromagnets driven by staggered spin–orbit fields

## Abstract

Within linear continuum theory, no magnetic texture can propagate faster than the maximum group velocity of the spin waves. Here, by atomistic spin dynamics simulations and supported by analytical theory, we report that a strongly non-linear transient regime due to the appearance of additional magnetic textures results in the breaking of the Lorentz translational invariance. This dynamical regime is akin to domain wall Walker-breakdown in ferromagnets and involves the nucleation of an antiferromagnetic domain wall pair. While one of the nucleated domain walls is accelerated beyond the magnonic limit, the remaining pair remains static. Under large spin–orbit fields, a cascade of multiple generation and recombination of domain walls are obtained. This result may clarify recent experiments on current pulse induced shattering of large domain structures into small fragmented domains and the subsequent slow recreation of large-scale domains.

## Introduction

Within a general physical context, a topological defect is characterised by a local/core region where order is highly altered whereas far from that core region the order parameter varies smoothly in space1. Topological defects can be found in a variety of fields such as in superfluid helium2 named vortices, in periodic crystal structures, they are called dislocations3,4,5 and in magnetism, magnetic domain walls (DWs)6,7, vortices8,9 and skyrmions10. In elementary dislocations, the underlying reason why these defects are called topological is because they cannot be made to disappear by a continuous deformation of the order parameter in infinite extended films. The measure of the distortion is associated to the so-called Burgers vector whose magnitude expresses the strength of the dislocation which is quantised to integer multiples of the atomic spacing of the lattice vectors11. Continuum-based textures such as magnetic textures like DWs, vortices and skyrmions are often characterised by the so-called winding number (a topological charge) whose value stands in direct relation to the magnitude of the normalised Burgers vector. However, topology is not at odds with instability. A relevant example of the previous is the plastic deformation and its role in dislocation dynamics12. Under low applied stresses, there is a quasilinear dependence between the dislocation velocity and the applied stress described by linear elasticity theory13. However, dislocation mobility is not restricted to subsonic regime under high strain rates. Surpassing of the sonic barrier was first postulated theoretically14,15,16 and then confirmed by atomistic simulations in bcc tungsten17,18 and in iron screw dislocations19 by means of introducing secondary kinks via mother–daughter nucleation process with opposite Burgers vector20.

Within the micromagnetic approximation, in a two-dimensional infinite magnetic system, the order parameter that breaks the symmetry is reflected by an average quantity: the magnetisation $$\overrightarrow{m}(x,y)$$ at each site in a given lattice. The connection between topology and magnetic media and its role in dynamics became apparent by the pioneering work of several authors21,22,23, when studying the dynamics of magnetic topological defects. The analytical approach consists of assuming the centre of the texture as a dynamical variable where the equation of motion can be derived using the Lagrangian formalism. Concretely, in ferromagnets, DWs are defined by the transition region that separates two homogeneously and opposite magnetised regions. In a similar fashion as for dislocations, a quantised topological invariant is defined for the magnetisation field24. By framing the magnetic texture on a closed path, it allows to define a map of the magnetisation configuration. Distinct topological textures are characterised by different winding number, w, which counts the number of times the magnetisation is wrapped onto itself25. The winding number density along the line at each point in time t, is $$w\left(x,t\right)=-{\nabla }_{x}\phi \left(x,t\right)$$ (with the total winding number, W, being $$\frac{1}{2\pi }\int\ w\left(x,t\right){ {d}}x$$, which for a 180° domain wall takes values $$\pm \frac{1}{2}$$). Here $$\phi \left(x,t\right)$$ is the in-plane angle of the spin at location x at time t of the spin-configuration along the 1D-line. In the case of Bloch walls separating magnetic domains in ferromagnets topological defects arise in the form of Néel lines26. Since the 1990s, the study of DW mobility in ferromagnets has been the flagship in Spintronics because of its potential in data storage27,28 and logic devices29. However, a major difficulty to implement such technological proposals arise as a result of the intrinsic instability the DW structure presents when it overcomes a certain threshold velocity. In this Article we show that this interesting feature of DW is present irrespective of whether the magnetic material is a ferromagnet or antiferromagnet. In ferromagnets, this regime is known as the Walker limit30 and it is characterised by a DW mobility that is the result of the combination of translational and oscillatory dynamics31 leading to periodic alteration of the DW chirality. Unfortunately, the Walker breakdown (WB) occurs at low domain wall velocities and imposes a limit on a reliable control of the DW motion. An alternative to prevent the WB consists of using nanotubes or nanocylinders32,33. These particular geometries have unveiled the spin analogue of the Cherenkov effect which involves the emission of electromagnetic waves by charge particles moving at speeds faster than the speed of light in the given medium34. In the case of antiferromagnets, several studies have provided significant insights on DW motion in antiferromagnets where high speeds have been predicted thanks to prevention of the WB35,36, yet there is an upper theoretical limit which cannot be surpassed. Interestingly, the manner in which the spin-space is conformed in antiferromagnets obeys the relativistic kinematics of special relativity where the analogue role played by the photons in conventional space is played by the magnons in a magnetic system. Consequently, by comparison, the ultimate limiting velocity of any texture is given by the maximum group velocity of the magnons, vg. In this work, we reveal that in analogue fashion as it occurs for dislocations under plastic deformation where the dislocations propagate at supersonic speeds, super-magnonic regime of motion can occur through a sequence of multiple nucleated textures mimicking the so-called WB.

## Results and discussion

To elucidate the existence of such type of magnetisation dynamics, we combined one-dimensional atomistic spin dynamics simulations of a layered antiferromagnet, Mn2Au37, with one-dimensional analytical theory. In order to induce magnetisation dynamics in Mn2Au (see Fig. 1a), we make use of the predicted staggered field-like torque in such crystal structures38, where the effective magnetic field resulting from a staggered-induced spin-density, $${\overrightarrow{H}}_{\text{so}}$$, possesses opposite signs at each sub-lattice and gives rise to a spin–orbit torque (see Fig. 1b). Some representative dynamics of the temporal evolution of DW position is obtained by extracting the spatiotemporal evolution of the winding number density shown in Fig. 1c, d. These spatiotemporal evolutions were obtained by firstly ramping the spin–orbit field (SO-field) from 0 to its maximum in a time interval of 10 ps and then it is kept constant till the end of the simulation Fig. 1e, f. The intrinsic inertial character of the translational motion of the DW is revealed by a non-linear trend that appears within the first 4–5 ps. The ballistic nature of the DW motion is revealed by the narrowing of the DW as it approaches vg according to the relativistic kinematics framework. The DW width at a given velocity is given by: $$\Delta (v)={(1-{v}^{2}/{v}_{\text{g}}^{2})}^{1/2}{\Delta }_{0}$$, where Δ0 = 19.8 nm is the DW width at rest. This contraction translates into an increase of the DW exchange energy. It is noteworthy to comment that in conventional special relativity theory, the contraction observed from non-inertial reference frame does not result in any incremental (or detrimental) interaction among the constituents of the object. First, the DW is accelerated to a speed that approaches to vg. After these transient dynamics, two distinct behaviours are obtained depending on the applied SO-field. For low values of SO-fields (see Fig. 1c), a saturation effect of the DW speed is observed, characterised by the slope of the maximum winding number density position (DW position), q, with respect to time, t with v = ∂tq. A linear variation of the DW speed as a function of the SO-field is consistent with previous reports36. In such a case, the energy dissipation induced by the viscous damping inhibits the DW to move faster as reported in previous work35. In contrast, at an onset of 65 mT, a different dynamical behaviour is observed (see Fig. 1d) which can be defined as the critical stress at which the DW suffers a threshold deformation which leads to the breeding of a DW-pair with trivial composite winding number, w, preserving the overall winding number (see Fig. 1f) . This strongly nonlinear regime which involves the generation of additional particles has been reported under several conditions in ferromagnets, for instance, domain wall WB mediated by the vortex–antivortex generation33, vortex-core reversal in spin-torque oscillators in nanodots39 and nanocontacts40. However, its occurrence in antiferromagnets has never been reported nor proposed. In ferromagnets, the physical origin for the appearance of new magnetic textures, concretely in the case of vortex-core reversal has been associated to the emergence of a effective magnetic field whose origin is purely dynamical and can be derived from the kinetic part of the Lagrangian density, $${\mathcal{L}}={{\mathcal{L}}}_{\text{kin}}-V$$. In our system, as the DWs are located in the ferromagnetic basal atomic planes where the coupling between the adjacent planes is antiferromagnetic one can derive the temporal evolution of the kinetic field using the Lagrangian formalism

$${{\vec{h}}}_{\text{kin}}(t)=\frac{1}{{M}_{\text{s}}}\frac{\delta {{\mathcal{L}}}_{\text{kin}}}{\delta {\vec{m}}(t)},$$
(1)

where Ms is the sub-lattice magnetisation and is given by 4μB/2V where μB is the Bohr magneton and V is the unit cell volume. Although this emergent field has already been proposed to explain the reversal of the vortex core polarity in ferromagnets where the dynamics are gyrotropic, we use this formalism also to translational motion of DWs in layered antiferromagnets. In antiferromagnets due to the relativistic nature of the dynamics, the DW suffers a substantial deformation due to its translational velocity. The extension and the kinetic field profile with respect to the moving DW depends upon the dynamical regime the DW is at a given instant (see Fig. 2a–d). During the first 4–5 ps of ultra-fast acceleration, the kinetic field is located behind the DW and smoothly transitions from the back to the centre and then to the forefront of the DW while mimicking the DW profile. The maximum component of the kinetic field is found to be along the $$\hat{z}$$-direction and can reach values up ~35 T in absolute value (see Supplementary Note 1). Contrary to ferromagnets where the magnetic texture absorbs energy from the kinetic field, we attribute the nucleation of DW pairs to the torque exerted by the kinetic field onto the local magnetisation promoting a reverse magnetic domain along the −x with an approximate extension just before the DW-pair nucleation of the exchange length, lex = $${(A/{K}_{2\parallel })}^{1/2}$$, where A is the effective exchange stiffness per unit area and K2 is the uniaxial anisotropy (see Table 1). The evolution of this reverse magnetic domain is governed by the competition between the exchange and anisotropy energies which results into breaking the reverse magnetic domain into a DW-pair with each of its constituents holding an opposite winding number (thus the sum of their windings is zero). We identify the time involved in the nucleation to be around 1 ps. Different types of magnetic distribution could in principle appear obeying the boundary conditions over the generated magnetic domain, however, we observed that the initially moving DW and the nucleated DW located just in front share the same winding number. The origin of this arrangement can be attributed to the minimisation of the total Zeeman energy now comprised of the SO-field and the kinetic field (see Supplementary Note 1). Therefore, while the torque provided by the kinetic field leads to a reverse magnetic domain in front of the moving DW with the appropriate extension to accommodate two DWs, the competition between the magnetic energies and the Zeeman energy gives rise to the generation of a DW-pair with a specific ordering.

The temporal evolution of the system after the DW-pair is present in the system is dramatically affected. Hereafter, we call the initially moving DW, DW1 and the nucleated DWs, DW2 and DW3 with DW2 always being the one closest to DW1 and DW3 located in front of DW2. As can be seen in Fig. 3a, immediately after the nucleation, DW1 and DW2 perform a few oscillations and then get stuck evolving in time but not in space. As DW1 and DW2 have the same winding number, they repel each other due to the ferromagnetic exchange interaction (see Fig. 3a). However, the region in between the pair (DW1 and DW2) has opposite orientation with respect to Hso and therefore, in order to reduce the Zeeman energy provided by the SO-field (note that the kinetic field is zero as the DWs are static), the distance between DW1 and DW2 has to be minimised. Consequently, as a result of the competition between the two forces, a local minimum appears that corresponds to a certain distance between the DW-pair. We verified this interpretation by calculating the energy barrier related to the distance among DW1 and DW2 (see Fig. 3b). The energy minimum corresponds to a given separation between the DWs’ centres, which depends on the magnitude of the SO-field (see Supplementary Note 2). From numerical simulations, we obtain a stable distance of 32 nm, whereas from analytical calculation the distance is 29 nm for Hso = 65 mT. Recent estimations of the effect of the current to produce a staggered local spin accumulations with opposite polarities are 2 mT per 107 A/cm2.

Another observation is that DW3 propagates forward by undergoing a velocity boost that surpasses the maximum group velocity of the magnons. This regime is transient and lasts for several tens of picoseconds which is sufficient for DW3 to cover over three microns. This mobility regime can be explained as follows. While the DW-pair is nucleated, DW1 and DW2 repel each other thanks to the ferromagnetic exchange interaction between them leading to a potential motion in opposite directions for each DW. Despite this fact, they barely move (see inset Fig. 3a) and therefore, all this stored energy is transferred from DW1 to DW3 in a similar fashion as reported by Tveten et al.41, where spin waves were able to transfer linear momentum to a given domain wall. Hence, the DW3 mobility at the moment of the nucleation consist of the velocity provided by the DW1, in addition to the boost coming from the momentum transfer due to the repulsion between DW1 and DW2. Using a collective coordinates approach combined with the Rayleigh dissipation function and the Euler–Lagrange equation of motion (see Supplementary Note 3), we obtain the DW3 velocity as a function of its distance with respect to the DW-pair to be

$${v}_{3} = \frac{\gamma }{\alpha }{H}_{\text{so}}{\Delta }_{1}\\ \,\,\,\, +\frac{2\gamma A}{\alpha {M}_{\text{s}}{\Delta }_{{\text{DW}}_{3}}^{2}}\left[x(t)\coth \left(\frac{x(t)}{{\Delta }_{{\text{DW}}_{3}}}\right)-{\Delta }_{{\text{DW}}_{3}}\right]{\rm{csch}}\left(\frac{x(t)}{{\Delta }_{{\text{DW}}_{3}}}\right),$$
(2)

where α is the Gilbert damping, $${\Delta }_{{\text{DW}}_{1}}$$ and $${\Delta }_{{\text{DW}}_{3}}$$ are the DWs’ width for DW1 and DW3 at the nucleation extracted from the atomistic simulations, respectively, and x(t) is the distance between DW1 and DW3. From numerical simulations, the extracted distance between DW1 and DW3 is 17 nm when the nucleation starts occurring and the extracted speed value for DW3 is around 177 km/s (see Fig. 3c) which is circa 4 times the maximum group velocity that magnons can theoretically attain in Mn2Au (see Supplementary Note 4). The maximum speed according to Eq. (2) is circa 133 km/s which corresponds to a distance of 16.1 nm between the DW1 and DW3. It is appealing to think that the special theory of relativity is being violated as no magnetic texture is by principle allowed to propagate faster than the maximum velocity of the magnons in the spin space. However, as soon as the DW-pair (DW2 and DW3) is present in the system additional interactions need to be taken into account which result into a Lagrangian that is not Lorentz invariant and therefore, the constraint on the maximum speed must be lifted up. Besides, when DW3 enters into a super-magnonic regime of motion, there is a sudden emission of spin-waves propagating together but never surpassing DW3 (see Fig. 2d). Analogous effect has been reported in mechanical systems such as in edge dislocation dynamics under shear stress where spontaneous emission of radiation has also been observed42,43. We attribute the origin of the emitted spin-waves to a mixture of the so-called Bremsstrahlung effect44 (or breaking radiation) and the spin-Cherenkov effect45. The Bremsstrahlung effect arises due to the deceleration of a charge particle. Therefore, one could speculate that such staggering deceleration could lead to excitations of the spin medium. As shown in Fig. 3d, DW3 not only surpass the maximum group velocity of the magnons but it also enters in a mobility regime where it moves faster than the phase velocity of the magnons. It is difficult in our case to clearly separate the existence of Bremsstrahlung and spin Cherenkov radiation. We note in Figs. 1d and 3a that the number of oscillations of DW1 and DW2 equals the number of spin-wave ripples travelling with DW3 and further, that the decay of these ripples appear to stretch over a rather long time. Therefore, one possibility for co-existence of both types of radiation could be that the spin-Cherenkov-originated spin waves acts in an anti-damping fashion onto the Bremsstrahlung in conjunction with the observation that the radiative ripples do not propagate in opposite direction to DW3. It is pertinent to explore how the system reacts when it has at its disposal additional pumped energy. We start by injecting an electric current with the time profile illustrated in Fig. 4a. The electric current has a rising time of 5 ps up to a peak value of Hso = 100 mT. The value of Hso is kept constant for the following 50 ps before reducing it to zero, with a falling time of 5 ps. The SO-field is set to zero for 50 ps before starting this pattern again three times in succession. The closed blue circles in Fig. 4a represents the maximum π/2-windings, i.e. the number of DWs in each pulse. In each cycle, we can observe an avalanche of DW-pairs. Once the primal DW nucleates the first DW-pair, the DW that propagates from the DW-pair (DW1 and DW2) at supermagnonic speeds becomes a new breeder and gives rise to new DW-pair. This phenomenon repeats for a 13 times leading to the appearance of 26 additional DWs in the system preserving at all times the overall topological charge. It is worth noticing that the DWs once nucleated do not reorganise into a more stable configuration while the SO-field is maximum giving rise to DW-lattice-like structure whose inter-DW distance depends upon the SO-field pulse pattern. The average DW density per pulse is 0.013 DW/nm and it takes ~20 ps to generate the whole DW lattice. Here closed orange circles denotes the number of DWs when the SO-field starts decreasing. Once this occurs, the lattice decompresses which, in combination to the attractive exchange interaction among DWs with opposite winding number, leads to recombination of some DWs. We observed a similar value for closed orange circles with pulse number meaning that the same number of DWs recombine once the SO-field reaches 0 mT.  The closed green circles represent the number of DWs extracted once the SO-field is about to rise up again. It can be seen from Fig. 4a that only spatially close situated DWs of opposite winding numbers annihilate increasing closed green circles in each pulse. This implies that the accumulated number of remnant DWs in the system after each pulse increases meaning that not all the generated DWs annihilate. After the fourth pulse, the number of DWs in the system is 8 with a spacing among the DWs that ranges from hundreds of nanometres to few micrometers. Initially the system contained two magnetic domains (separate by a single DW) and after four pulses the number of magnetic domains Nm, has increased by Nm = Ndws + 1, where, Ndws, accounts for the number of DWs in the system. More complex DW-crystal lattices can be obtained by varying the pulse duration and the sign of the SO-field (see Supplementary Note 5) showing the rich magnetisation dynamics in these type of systems. We wish at this point to also stress some observations (not shown here), that there appears to be no clear correlation between the strength of Hso and the number of DWs nucleated as well as the point in time and space where they are nucleated. This opens up the possibility that this system under the current excitation circumstances could potentially exhibit chaotic behaviour. This will be the topic of future work however. We note at this stage, that recent work on current-induced resistance changes in the antiferromagnet CuMnAs was, in conjunction with imaging attributed to the fragmentation and recovery of the domain structure46,47. In particular they observed a gradual increase in the resistance with repeated pulsing and a slow relaxation of the resistance towards lower values than after the last pulse when turning off the excitation completely. However, provided that finite size-effects can be excluded and that such a system observes global conservation of the total winding number, it should be possible to return to the original state by applying a reverse field below the AFM Walker-breakdown field, in order to recover completely the initial state of resistance, in other words, a complete reset of the system. Further, it is noteworthy that the relaxed resistance value after a long waiting time was higher than the original starting resistance. This could mean that residual DW are present. If the interpretation that the observed resistance changes is due to domain fragmentation and recovery/recombination is correct we cannot help but to speculate about possible measurements of our system under study. Although the mechanism of domain fragmentation in our case is due to the AFM WB in a perfect crystal and the results in the reported experimental work likely has heat as the main cause of fragmentation, this means that that such resistance variations in analogous systems where pinning and heating effects can be ruled out could be used as an indirect detection method for the occurrence of the AFM WB process and its associated generation of supermagnonic textures. It must be pointed out that for this particular pulse ramp-time no signature of DW-pair nucleation is obtained well after the SO-field has reached Hcrit = 65 mT, meaning that the critical feld depends upon the ramping time conditions rather than the absolute value of the energy pumped into the system. Figure 4b shows the transition from a system with a DW-lattice with 13 DW-pairs towards a system with only one DW-pair due to the multiple recombination when the SO-field is zero. It is noted that there is a distribution of distances that separate consecutive DWs with opposite winding number. This gives rise to a manifold of recombination times as shown in Fig. 4c. Analytically, it is possible to obtain the recombination time as a function of the distance by assuming that the decompression introduced by the absence of SO-field results in a transition phase from a DW-lattice to a DW-gas for a transient time. Therefore, the recombination time of a given DW-pair is only governed by intrinsic parameters such as, the exchange interaction, the damping and the distance between the DWs ruling out the effect of the presence of the rest of DW-pairs in the system. We can derive the recombination time from a general expression of the exchange interaction between two domain wall pairs with opposite winding number as

$${\tau }_{\mathrm{r}}(l)=\int_{{t}_{0}}^{{t}_{\mathrm{r}}}{\mathrm{{d}}}t=-\frac{\alpha {{M}_{\mathrm{s}}}\Delta^{2} }{2\gamma A}\int_{l}^{{l}_{0}}{ {d}}x\frac{1}{\left[\Delta -x{\mathrm{coth}} \left(\frac{x}{\Delta }\right)\right]{\mathrm{csch}}\left(\frac{x}{\Delta }\right)},$$
(3)

where Δ is the DW-width of each DW, l represents the distance each DW needs to travel before the recombination occurs at t0, l0 = 1 nm and tr represents the recombination time. We note that the predicted recombination time reproduce quantitatively the recombination time values extracted from the atomistic spin dynamic simulations which validates the hypothesis of 1D DW-pairwise gas approximation. For distances in the range of few hundreds of nanometres the recombination time is in the range of few picoseconds as the exchange interaction is a short-range interaction. However, we can see from Fig. 4b that there are DWs remaining in the system whose separation is in the range of few microns. For a given distance of 1.8 μm, the expected recombination time lies in the range of  ~6 days. This suggests that in an ideal scenario where pinning effects, thermal fluctuation effects are not present, the stability of such a configuration is granted thanks to the absence of long-range interaction. We can observe in Fig. 4b that once the SO-field is set to zero, each of the recombination processes produces an excitation in the continuum spectrum. This highly nonlinear process yields to a deformation amplitude that oscillates in time with a very precise frequency. The emergence of this breather-like excitation is due to the excess of kinetic energy carried by each DW involved in the collision whose value is not large enough to escape the attractive potential provided by its anti-particle. By mapping the breather into a simple damped harmonic oscillator, we obtained a good quantitative agreement on the breather lifetime (see Supplementary Note 6). The breather decay occurs in 20–30 ps and it is governed by the Gilbert damping. Moreover, the characteristic frequency of the breather is determined to be 544 GHz frequency, which lies within the linewidth of the calculated spin-wave band gap at Hso = 100 mT, i.e. the frequency at zero wave vector, k = 0. We are thus compelled to assign the breather frequency to that of the band-gap.

## Methods

We perform numerical simulations via atomistic spin dynamics simulations. For this, the full Mn2Au crystal structure is taken into consideration. Two formula units as that in Fig. 1a in the main text is stacked on top of each and then replicated along the x-direction 60,000 times. The system has open boundaries along x and z while periodic boundary conditions are imposed along y. The time evolution of a unit vector spin at site i, Si, is simulated by solving the Landau–Lifshitz–Gilbert equation:

$$\frac{{\rm{{d}}}{{\bf{S}}}_{i}}{{\rm{{d}}}t}=-\gamma \ {{\bf{S}}}_{i}\times {{\bf{H}}}_{i}^{\,\text{eff}\,}-\gamma \alpha \ {{\bf{S}}}_{i}\times \left({{\bf{S}}}_{i}\times {{\bf{H}}}_{i}^{\,\text{eff}\,}\right),$$
(4)

where γ is the gyromagnetic ratio of a free electron (2.21 × 105 m/A s), α is the Gilbert damping set here to 0.001 and $${{\bf{H}}}_{i}^{\,\text{eff}\,}$$ is the effective field resulting from all of the interaction energies. The energies taken into account are the three exchange interactions (two antiferromagnetic and one ferromagnetic), magneto crystalline energy contributions and the spin–orbit field. The total energy, E, considered is

$$E= -2\sum _{\langle i< j\rangle }{J}_{ij}{{\bf{S}}}_{i}\cdot {{\bf{S}}}_{j}-{K}_{2\perp }\sum _{i}{\left({{\bf{S}}}_{i}\cdot \hat{{\bf{z}}}\right)}^{2}-{K}_{2\parallel }\sum _{i}{\left({{\bf{S}}}_{i}\cdot \hat{{\bf{y}}}\right)}^{2}\\ - \frac{{K}_{4\perp }}{2}\sum _{i}{\left({{\bf{S}}}_{i}\cdot \hat{{\bf{z}}}\right)}^{4}-\frac{{K}_{4\parallel }}{2}\sum _{i}\left[{\left({{\bf{S}}}_{i}\cdot {\hat{{\bf{u}}}}_{1}\right)}^{4}+{\left({{\bf{S}}}_{i}\cdot {\hat{{\bf{u}}}}_{2}\right)}^{4}\right]-{\mu }_{0}{\mu }_{\rm{{{s}}}}\sum _{i}{{\bf{S}}}_{i}\cdot {{\bf{H}}}_{i}^{\,\text{eff}\,}.$$
(5)

The first term on the right-hand side is the exchange energy where Jij is the exchange coefficient along the considered bonds (see Supplementary Note 4). The second and third terms are the uniaxial hard and easy anisotropies of strengths K2 and K2, respectively, while the fourth and fifth terms collectively describes tetragonal anisotropy. For the in-plane part of the tetragonal anisotropy, u1 = $$\left[110\right]$$ and u2 = $$\left[1\bar{1}0\right]$$. Finally, μ0 and μs are the magnetic permeability in vacuum and the magnetic moment, respectively. We have used μs = 4μB, with μB being the Bohr magneton. It is to be noted that the tetragonal anisotropy was included for sake of completeness as it is present in this material but its role in the high speed dynamics is negligible due to the weak magnitude of its anisotropy constants. The effective field is then subsequently evaluated at each point in time using Eq. (5) as $${{\bf{H}}}_{i}^{\,\text{eff}\,}=\frac{-1}{{\mu }_{0}{\mu }_{\rm{{{s}}}}}\frac{\delta E}{\delta {{\bf{S}}}_{i}}$$. The system of equations, Eq. (4) are solved by a fifth-order Runge–Kutta Method. Spatio-temporal data is analysed on an extracted 1-dimensional line of the computational domain (see Supplementary Note 4). Material constants used are summarised in Table 1.

## Data availability

The data generated for the current study are available from the corresponding author upon reasonable request.

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Tatara, G., Kohno, H. & Shibata, J. Microscopic approach to current-driven domain wall dynamics. Phys. Rep. 468, 213–301 (2008).

29. 29.

Allwood, D. et al. Magnetic domain-wall logic. Science 309, 1688–1692 (2005).

30. 30.

Schryer, N. L. & Walker, L. R. The motion of 180 domain walls in uniform dc magnetic fields. J. Appl. Phys. 45, 5406–5421 (1974).

31. 31.

Mougin, A. et al. Domain wall mobility, stability and Walker breakdown in magnetic nanowires. Europhys. Lett. 78, 57007 (2007).

32. 32.

Hertel, R. & Kirschner, J. Magnetization reversal dynamics in nickel nanowires. Physica B 343, 206–210 (2004).

33. 33.

Yang, M. et al. Magnetization reversal dynamics in nickel nanowires. Appl. Phys. Lett. 99, 122505 (2011).

34. 34.

Čerenkov, P. A. Visible radiation produced by electrons moving in a medium with velocities exceeding that of light. Phys. Rev. 52, 378 (1937).

35. 35.

Shiino, T. et al. Antiferromagnetic domain wall motion driven by spin–orbit torques. Phys. Rev. Lett. 117, 087203 (2016).

36. 36.

Gomonay, O., Jungwirth, T. & Sinova, J. High antiferromagnetic domain wall velocity induced by Néel spin–orbit torques. Phys. Rev. Lett. 117, 017202 (2016).

37. 37.

Roy, P. E., Otxoa, R. M. & Wunderlich, J. Robust picosecond writing of a layered antiferromagnet by staggered spin–orbit fields. Phys. Rev. B 94, 014439 (2016).

38. 38.

Železný, J. et al. Relativistic Néel-order fields induced by electrical current in antiferromagnets. Phys. Rev. Lett. 113, 157201 (2014).

39. 39.

Guslienko, K. et al. Dynamic origin of vortex core switching in soft magnetic nanodots. Phys. Rev. Lett. 100, 027203 (2008).

40. 40.

Watelot, S. P. et al. Commensurability and chaos in magnetic vortex oscillations. Nat. Phys. 8, 682 (2012).

41. 41.

Tveten, E. G., Qaiumzade, A. & Brataas, A. Antiferromagnetic domain wall motion induced by spin waves. Phys. Rev. Lett. 112, 147204 (2014).

42. 42.

Peng, S. et al. Supersonic screw dislocations gliding at the shear wave speed. Phys. Rev. Lett. 122, 045501 (2019).

43. 43.

Nosenko, V., Zhdanov, S. & Morfill, G. Supersonic dislocations observed in a plasma crystal. Phys. Rev. Lett. 99, 025002 (2007).

44. 44.

Tesla, N. An interesting feature of X-ray radiations. Electron. Rev. 29, 13–14 (1896).

45. 45.

Yang, M. Spin-Cherenkov effect and magnonic Mach cones. Phys. Rev. B 88, 220412 (2013).

46. 46.

Wörnle, M. S. et al. Current-induced fragmentation of antiferromagnetic domains. arxiv: 1912.05287 (2019).

47. 47.

Kăspar, Z. et al. High electrical and optical resistive switching in a thin film antiferromagnet. arxiv: 1909.09071 (2019).

## Acknowledgements

The work of R.M.O. and K.Y.G. was partially supported by the STSM Grants from the COST Action CA17123 “Ultrafast opto-magneto-electronics for non-dissipative information technology”. K.Y.G. acknowledges support by IKERBASQUE (the Basque Foundation for Science) and Spanish MINECO project FIS2016-78591-C3-3-R.

## Author information

Authors

### Contributions

J.W., P.E.R. and R.M.O. initiated the project. P.E.R. developed the atomistic code. R.M.O. and P.E.R. analysed the data and interpreted the simulations result. R.M.O., K.Y.G. and R.R.-E. developed the analytical model. R.M.O., P.E.R., R.R.-E., J.G., K.Y.G. and J.W. discussed the data. R.M.O. wrote the manuscript.

### Corresponding author

Correspondence to Rubén M. Otxoa.

## Ethics declarations

### Competing interests

The authors declare no competing interests.

Publisher’s note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

## Rights and permissions

Reprints and Permissions

Otxoa, R.M., Roy, P.E., Rama-Eiroa, R. et al. Walker-like domain wall breakdown in layered antiferromagnets driven by staggered spin–orbit fields. Commun Phys 3, 190 (2020). https://doi.org/10.1038/s42005-020-00456-5

• Accepted:

• Published:

• ### Magnon pair emission from a relativistic domain wall in antiferromagnets

• Gen Tatara
• , Collins Ashu Akosa
•  & Rubén M. Otxoa de Zuazola

Physical Review Research (2020)


text:
# Optionally enforce URL-safe domain and project names¶

bp url-safe-naming

In preparation for future support for hierarchical naming of projects and domains, deprecate non url-safe names as well as optionally enforce url-safe names.

## Problem Description¶

Once upon a time, in a release far away, there were just tenants in a flat namespace. It was natural that tenant names had to be unique keystone-wide. Subsequently we have added domains (to hold tenants, now called projects), which meant that project names had only to be unique within that domain. We now support hierarchies of projects (within a domain) and potentially may support hierarchies of domains above projects. However, our name uniqueness rules have not changed, so a project name must (effectively) be unique within its hierarchy, and domain names must still be unique keystone-wide. One main reason our rules have not changed is that we support scoping to domain and project by name.

Given this new hierarchy support, our naming rules will become increasingly onerous on customers. Not only for customer with deep project hierarchies, but also will complicate any support for domain hierarchies for resellers. In the future it would be ideal if you could address a node in a hierarchy with a url-style addressing scheme, for example:

{
"auth": {
"identity": {
...
},
"scope": {
"project": {
"domain": {
"name": "acme.com"
},
"name": "development/sas/myproject"
}
}
}
}

The same addressing scheme could also be used for domains, if we require support for domain hierarchies.

However, our current project and domain names are not url-safe, so we could not guarantee to support any particular hierarchical naming scheme without risk of breaking access to strangely named projects/domains.

## Proposed Change¶

It is proposed that we lay the groundwork to move to url-safe naming for all projects and domains. A url-safe name is defined as one that does not contain any of the reserved characters as defined in section 2.2 of rfc3986 <http://tools.ietf.org/html/rfc3986>_.

In particular, we would:

• Deprecate url-unsafe naming, so deprecation warnings would be logged for unsafe names.
• Provide a keystone-manage option to list the ID and name of any url-unsafe projects or domains, so that such names can subsequently be updated using the regular Identity API.
• Provide two configuration options (one for projects and one for domains) to enable enforcement. These would accept three values: off, new, strict. These would be off by default, which means no change in behavior.
• If set to new then a request to create a new domain or project with a name that does not match the url safe criteria will be rejected.
• If the configuration option is set to strict, in addition to the restrictions defined for new, any existing unsafely-named domains or projects will be treated as disabled for the purposes of issuing a token scoped by project/domain name. Renaming the project or domain will re-enable it.

Future keystone releases would harden the above, moving towards a situation when the configuration options are eventually deprecated and url-safe naming is permanently enabled.

New features we release, for example domain hierarchies for resellers, could insist that the appropriate url-safe configuration option was enabled before that feature was supported.

To be clear, this proposal does not enable the actual use of hierarchical naming, rather it lays the groundwork for us to do so in the future.

This proposal does not preclude the use of unicode for project and domain names (since all the reserved characters are coded the same in ascii and unicode). However, to utilize such an extended character set for auth or searching by name would require additional changes (for encoding and decoding) which is outside of the scope of this proposal. By allowing unicode within the strict naming option defined here, we allow customer using unicode to “go strict” as soon as they can (and hence avoiding a further road block to us being able to deprecate unsafe names).

None

None

None

None

None

None

None

### Other Deployer Impact¶

None, by default, although would require them to potentially rename projects or domains if they want to start enforcing url-safe naming.

None

## Implementation¶

### Assignee(s)¶

Primary assignee:
henry-nash

### Work Items¶

• Add support for new configuration options
• Add checking in create/update domain/project
• Add checking in auth to prevent scoping to unsafe entities
• Add support to keystone-manage for listing unsafely-named domains/projects

None

## Testing¶

None, above and beyond unit testing

## Documentation Impact¶

Changes to the Identity API to clarify naming.

None

format: plain


text:
Nonstandard Analysis models using non-well-founded sets

1. Feb 13, 2008

Dragonfall

I've been told that non-well-founded sets can be used to model nonstandard analysis, but I can't find any literature on the subject. Does anyone know where I can find it?

2. Feb 13, 2008

Hurkyl

Staff Emeritus
My best guess at what you mean is the following...

First see this section of Wikipedia's article on NSA.

In order to do "ordinary" NSA, no non-well-foundedness is needed. Note that what they denote as V is actually $V_\omega$.

As I understand it, a problem occurs only when you consider unbounded formulae, or if you want to work in some $V_\alpha$ with $\alpha > \omega$. In order to do either of these, you must assume some version of anti-foundation.

Last edited: Feb 13, 2008
3. Feb 14, 2008

gulliput

see section 8.3 'Non-well-founded set theories' in V.Kanovei and M.Reeken's book 'Nonstandard analysis, axiomatically' for a review.

you might also be interested in our Theory of Hyperfinite Sets where 'nonstandard analysis' is being developed within a non-well founded universe of hereditarily hyperfinite sets:
http://arxiv.org/abs/math/0502393

it is also interesting that (roughly speaking) what is needed for nonstandard analysis is infinitely descending 'membership chains', no 'cyclic memberships' are needed (although non-well-foundedness is usually associated with the possibility of cycles)

4. Feb 14, 2008

gulliput

Wikipedia article's description of logical foundations of NSA is quite limited and also outdated, to my opinion.

Every nonstandard model IS non-wellfounded "in the standard sense" - if it is countably saturated ("a minimal requirement" for doing NSA) then there are infinitely descending 'membership chains' - $x_1\ni x2 \ni \ldots$

The nonstandard $V_\omega$ is "actually" a substructure of a saturated elementary extension of the standard $V_\omega$ - therefore it is non-wellfounded.

5. Feb 14, 2008

Dragonfall

Thanks for the replies. I'll check out the book.

6. Feb 14, 2008

Hurkyl

Staff Emeritus
There's no problem with $x_1$ being untyped?

Or, I suppose a more accurate question is whether requiring constants to be typed eliminates this issue.

Last edited: Feb 14, 2008
7. Feb 14, 2008

gulliput

I was inaccurate stating that in the "superstructure NSA" (the framework described in the Wikipedia article) there is non-wellfoundedness. The superstructure is indeed a substructure of an elementary saturated extension of some standard structure but this substructure is built so that it eliminates non-wellfoundedness of the extension. But this has its issues: for instance, you cannot have von Neumann natural numbers (every n is the set of smaller numbers) "on the ground floor" if you want no nonwellfoundedness whatsoever.

What is important for me about nonwellfoundedness is that it allows to consider the universe (of sets) being genuinely nonstandard (we were not aware of its nonstandardness before due to our limitations) - instead of seeing the nonstandard universe as some weird structure on top and inside of "the true universe".

8. Feb 14, 2008

Dragonfall

I've read the section from Wikipedia and I can't see why we should evoke non-well-founded sets or use V_a for a greater than omega.

9. Feb 17, 2008

gulliput

as i said the contents of this section in Wikipedia is much narrower than it's title (it is not encyclopedic enough) since it does not speak bout more recent work on axiomatic foundations of nsa.

answering to your question - if we are satisfied with superstructures - we don't have to. but we might, and if we would we could feel better. we would have another picture, and a picture does make a difference. there is non-well foundedness in every non-standard 'full-scale' axiomatic set-theoretical framework i know of. but we don't have to use axiomatic frameworks! but we don't have to use nsa either! :)


text:
# VietNam 2013

August 4-10, 2013

Nanophysics: from fundamentals to applications

(the return)

Thursday 8
Dots and wires

› 14:00 - 14:20 (20min)
Two-impurity Kondo Effect in Al/AlOx/Y Tunnel Junctions
Yu-Ren Lai  1, *@  , Da-Wei Wang  1@  , Juhn-Jong Lin  1, 2, *@
1 : Institute of Physics , National Chiao Tung University
1001 University Road, Hsinchu 300 -  Taiwan
2 : Department of Electrophysics, National Chiao Tung University
1001 University Road, Hsinchu 300 -  Taiwan
* : Corresponding author

We have fabricated a series of Al/AlOx/Y tunnel junctions and measured the differential conductance G(V; T) at liquid-helium temperatures. We found that the zero-bias conductance G(0; T) increases with reducing T below 40 K, i.e., G(0; T) obeys a -ln T law at a higher T regime (10-25 K) and crosses over to a -$\sqrt{T}$ law at an intermediate T regime (5-20 K). The unique -$\sqrt{T}$ feature is suggestive of a novel Kondo effect. In particular, in this intermediate T regime, we observed that the finite-bias G(V; T) curves at different T's collapse closely with the universal scaling function of the two-impurity Kondo effect.

Furthermore, while the G(V; T) behavior is essentially similar for all junctions at not too low T, we found two kinds of distinct T dependences of G(0; T) at T

The two-impurity Kondo effect arises from a minute number of spin-1/2 yttrium impurities which diffused into the insulating barrier during the junction fabrication process.

 Online user: 1 RSS Feed


text:
# Physics equations/07-Work and Energy/A:pendulum

## Wikilab: Energy and the pendulum

By definition, angular frequency and period are related by,

${\displaystyle \omega T=2\pi }$

This lab employs two approximation that are valid when ${\displaystyle \theta <<1}$:

${\displaystyle \sin \theta \approx \theta \qquad \cos \theta \approx 1-{\frac {1}{2}}\theta ^{2}}$

Use the Excel spreadsheet to plot the sine and cosine functions for θ between zero and one, taking approximately 20 increments within that interval.

Copy and paste your page onto the same excel spreadsheet and verify that a better approximation is:

${\displaystyle \sin \theta \approx \theta -{\frac {\theta ^{3}}{3\cdot 2}}\qquad \cos \theta \approx 1-{\frac {1}{2}}\theta ^{2}+{\frac {\theta ^{4}}{4!}}}$

Consider a pendulum of length L situated so that the equilibrium point is at the origin. Make a careful hand drawing showing that:

${\displaystyle x\approx L\theta \qquad \mathrm {and} \qquad y\approx {\frac {1}{2}}L\theta ^{2}}$

Use this to show that an approximate formula for the energy of a pendulum is:

${\displaystyle E={\frac {1}{2}}mv^{2}+{\frac {mg}{2L}}x^{2}}$

Defining ${\displaystyle k=mg/L}$ as the effective spring constant, this energy can be expressed in familiar form:

${\displaystyle E={\frac {1}{2}}mv^{2}+{\frac {1}{2}}kx^{2}}$


text:
Help

Image text transcribed for accessibility: STEP RESPONSE OF AN UNDE RLC NETWORK For the network below, calculate the Laplace Domain solution for the transfer function, Vout (s)/Vin , using a value of L = 4.0mH and L = 0.2 mu F, calculate the natural radian frequency omega n, and the Damping coefficients 5, in terms of R. From this equation, determine the value of for xi =0.7, 0.1, and 0.01, using these values of R and assuming a step input of 10V, solve for the time after the step when the output will reach the maximum and first minimum using the complete solution of Vout (t), determine the voltage at these maximum and minima value the output step is 10volts. STEP RESPONCE OF CRITICALL RLC NETWORK For the network below, calculate the Laplace domain solution for the transfer function, Vout (s)/Vin (s). Again using L = 4.0 mH and C = 0.2 mu F. Calculate and xi in terms of R. From these equations will result in a initially response.


text:
# Pivot partitioning by scanning

Algorithmic problem: Pivot partioning

Prerequisites: None

Type of algorithm: loop

Auxiliary data: Five index pointers, $m_1,m_2,i_1,i_2,i_3 \in \mathbb N.$

## Abstract View

Invariant:

1. The values of $m_1$ and $m_2$ only depend on the input sequence $S$ and are, hence, constant throughout the loop. More specifically, in $S$ there are
1. $m_1-1$ elements less than $p$,
2. $m_2-m_1$ elements equal to $p$, and
3. $n - m_2+1$ elements greater than $p$.
2. Before and after each iteration, it is:
1. $1 \le i_1 \le m_1 \le i_2 \le m_2 \le i_3 \le n+1$;
2. $S[j] \lt p$ for $j \in \{1,...,i_1-1\}$; if $i_1 \lt m_1$, it is $S[i_1] \nless p$;
3. $S[j] =p$ for $j \in\{m_1,...,i_2-1\}$; if $i_2 \lt m_2$, it is $S[i_2] \ne p$;
4. $S[j] \gt p$ for $j \in \{m_2,...,i_3-1\}$; if $i_3 \le n$, it is $S[i_3] \ngtr p$.

Variant: At least one of $i_1,i_2$, and $i_3$ is increased by at least $1$, none of them is decreased.

Break condition: It is $i_1=m_1, i_2=m_2$ and $i_3 = n+1$.

## Induction basis

Abstract view:

1. Set $m_1$ and $m_2$ according to the loop invariant.
2. Initialize $i_1,i_2$ and $i_3$.

Implementation:

1. Set $m_1:=1$ .
2. For $j \in {1,...,n}$: If $S[j]\lt p$ increase $m_1$ by $1$.
3. Set $m_2:=m_1$.
4. For $j \in {1,...,n}$: If $S[j]=p$ increase $m_2$ by $1$.
5. $i_1:=1$.
6. $i_2:=m_1$.
7. $i_3:=m_2$.
8. While $i_1\lt m_1$ and $S[i_1]\lt p$, set $i_1:=i_1+1$.
9. While $i_2\lt m_2$ and $S[i_2]=p$ , set $i_2:=i_2+1$.
10. While $i_3\lt n+1$ and $S[i_3]\gt p$, set $i_3:=i_3+1$ .

Proof: Obvious.

## Induction step

Abstract view:

1. If the algorithm is not done yet, identify two out of the three index pointers $i_1,i_2$, and $i_3$such that at least one of these two can be moved one step forward after a swap of the elements at these two positions.
2. Then move each of $i_1,i_2$, and $i_3$ forward as long as the invariant is not violated.

Implementation:

1. If $i_1=m_1$ and $i_2=m_2$ and $i_3=n+1$, terminate the algorithm and return $m_1,m_2$.
2. If $i_1\lt m_1$ :
1. If $S[i_1]=p$:
1. Swap $S[i_1]$ and $S[i_2]$.
2. Set $i_2:=i_2+1$ until $i_2=m_2$ or else $S[i_2] \ne p$.
2. Otherwise (that is, $S[i_1]\gt p$):
1. Swap $S[i_1]$ and $S[i_3]$.
2. Set $i_3:=i_3+1$ until $i_3=n+1$ or else $S[i_3] \ngtr p$.
3. While $i_1\lt m_1$ and $S[i_1]\lt p$: set $i_1:=i_1+1$.
3. Otherwise (that is, $i_1=m_1$):
1. Swap $S[i_2]$ and $S[i_3]$.
2. Set $i_2:=i_2+1$ until $i_2=m_2$ or else $S[i_2] \ne p$ .
3. Set $i_3:=i_3+1$ until $i_3=n+1$ or else $S[i_3] \ngtr p$.

Correctness: Due to the various "otherwise" clauses, all possible cases are covered. Therefore, it remains to show that the invariants are maintained in each of the considered cases. As the values of $m_1$ and $m_2$ are never changed after initialization, all three parts of the first invariant are trivially maintained. Since $i_1$ (resp., $i_2$, $i_3$) is only increased in case $i_1\lt m_1$ (resp., $i_2\lt m_2$, $i_3\lt n+1$), Invariant 2.1 is maintained as well. So, we will focus on Invariants 2.2-2.4. If Step 2 applies, maintenance of Invariants 2.2-2.4 might be obvious. So consider Step 3. If Step 3 applies ($i_1=m_1$), the break condition is not yet met, so we know $i_2\lt m_2$ or $i_3\lt n+1$, or both. Without loss of generality, assume $i_2\lt m_2$ (the other case is mirror-symmetric). Invariant 2.3 implies $S[i_2]\neq p$, which amounts to $S[i_2]\gt p$ in Step 3. So, not all elements $\gt p$ are on the correct positions, which implies $i_3\lt n+1$. In summary, it is safe to apply Step 3.

## Java Implementation

    //snipped from https://foo.algo.informatik.tu-darmstadt.de
//preperated for Spec
//AUTHOR: Thomas Lüdecke
//30.05.2015 05:22 | git 4072e171

private List<Integer> list executePivotPartitioning(int pivot, int m1, int m2,  List<Integer> list){
//init and save
int i1 = 0;
int i2 = m1;
int i3 = m2;

//int i = 0; //Iteration Counter
//Inductionbase
while(i1 < m1 && list.get(i1) < pivot){
i1++;
}

while(i2 < m2 && list.get(i2) == pivot){
i2++;
}

while(i3 <list.size() && list.get(i3) > pivot){
i3++;
}

//each Iteration is one induction step
while(i1 < m1 || i2 < m2 || i3 < list.size()) {

if(i1 < m1) {
if(list.get(i1) == pivot) {
swap(i1,i2, list);
while(i2 < m2 && list.get(i2) == pivot){
i2++;
}
}else{
swap(i1,i3, list);
while(i3 <list.size() && list.get(i3) > pivot){
i3++;
}
}
while(i1 < m1 && list.get(i1) < pivot){
i1++;
}
//i++; //iteration end
}else{
swap(i2,i3, list);
while(i2 < m2 && list.get(i2) == pivot){
i2++;
}
while(i3 <list.size() && list.get(i3) > pivot){
i3++;
}
//i++; //iteration end
}
}
return list;
}

//AUTHOR: Alexander Jandousek
private void swap(int i1, int i2, List<Integer> list){
int temp = list.get(i1);
list.set(i1, list.get(i2));
list.set(i2, temp);
}

//These methods are used to init m1 and m2 before calling executePivotPartitioning(...)
//AUTHOR: Alexander Jandousek
private int calcM1(int pivot,  List<Integer> list){
int m1 = 0;
for(int n : list){
m1 = (n < pivot)? m1+1: m1;
}
return m1;
}

//AUTHOR: Alexander Jandousek, Thomas Lüdecke
private int calcM2(int pivot,  List<Integer> list){
int m2 = calcM1(pivot,list); //not optimized
for(int n : list){
m2 = (n == pivot)? m2+1: m2;
}
return m2;
}


text:
# Multiscale stochastic volatility for variance swaps with constant elasticity of variance

Ji Su Yu, Jeong Hoon Kim

Research output: Contribution to journalArticlepeer-review

## Abstract

The variance swap is one of volatility derivatives popularly used for the risk management of financial instruments traded in volatile market. An appropriate choice of a volatility model is an important part of the risk management. One of desirable considerations should be given to the fact that the volatility varies on several characteristic time scales. Stochastic volatility models can reflect this feature by introducing multiscale volatility factors. However, pure stochastic volatility models cannot capture the whole volatility surface accurately although the model parameters have been calibrated to replicate the market implied volatility data for near at-the-money strikes. So, we choose a hybrid model of constant elasticity of variance type of local volatility and fast and slow scale stochastic volatility for evaluating the fair strikes of variance swaps. We obtain a closed-form solution formula for the approximate fair strike values of continuously sampled variance swaps and compute the solution. The theoretical formula is validated through Monte Carlo simulation. The predictability of the strike price movements is discussed in terms of the sensitive effects of the stochastic volatility and the elasticity of variance parameters for a given partial information about the underlying asset and volatility.

Original language English Soft Computing https://doi.org/10.1007/s00500-022-07679-4 Accepted/In press - 2022

### Bibliographical note

Funding Information:
The research of J.-H. Kim was supported by the National Research Foundation of Korea NRF2021R1A2C1004080.


text:
## IsTim Group Title Which pairs of planes are parallel and distinct and which are conincident? 2x+4y-7z-2=0 4x+6y-14z-8=0 2 years ago 2 years ago

1. IsTim Group Title

I tried looking through my notes and textbook for this one, but unless there is a direct connection, I don't understand how to apply it to this question.

2. IsTim Group Title

I managed to answer some unrelated questions beforehand, but I don't know how I managed to solve them...

3. IsTim Group Title

So I figured out the normals were [2,3,7] and [4,6,-14]

4. IsTim Group Title

I can see that I just have to multiply 2,3,7 by 2 to get the other plane. So would that be conidental or parallel and distinct?

5. bahrom7893 Group Title

that should be parallel and distinct.. if u can just multiply them by a scalar, but i suck at this, so idk.. @Mertsj @Zarkon

6. bahrom7893 Group Title

@amistre64 @satellite73 @myininaya

7. bahrom7893 Group Title

@LagrangeSon678

8. IsTim Group Title

Wait. Maybe I'll try something out here.

9. Zarkon Group Title

how is [2,3,7] the normal vector?

10. IsTim Group Title

Er, I just looked at my notes and inferred that...

11. Zarkon Group Title

if 2x+4y-7z-2=0 is your equation then <2,4,7> is the normal vector

12. Zarkon Group Title

<2,4,-7>

13. bahrom7893 Group Title

Zarkon to the rescue!!!!

14. IsTim Group Title

Ok, got that. For the other then, it must be 4,6,-14?

15. Zarkon Group Title

yes

16. IsTim Group Title

So from this, how would I determine the answer? Or do I need more?

17. LagrangeSon678 Group Title

if the planes are parrallel then the cross product of their normal vectors should be 0

18. Zarkon Group Title

two plane are parallel iff their normal vectors are parellel

19. IsTim Group Title

I'll try that cross product thing. Thanks guys. I'lll cry out if I have a problem.

20. Zarkon Group Title

two non zero vectors $$\vec{v}$$ and $$\vec{u}$$ are parallel iff $$\vec{v}=c\vec{u}$$ for some scalar $$c$$

21. IsTim Group Title

Pardon?

22. Zarkon Group Title

is one of the vectors a constant multiple of the other?

23. IsTim Group Title

The first one, multiplied by 2, equals the second one.

24. Zarkon Group Title

really?

25. IsTim Group Title

IT seemed so...

26. Zarkon Group Title

if you have the equations 2x+4y-7z-2=0 4x+6y-14z-8=0 then you have the normal vectors <2,4,-7> and <4,6,-14> are these constant multiples of each other?

27. IsTim Group Title

That 4y was suppose to be a 3y....

28. IsTim Group Title

I did mistype that. No wonder my confusion. And probably yours...

29. Zarkon Group Title

then <4,6,-14>=2<2,3,-7> and so they are parallel...and therefore their planes are parallel

30. IsTim Group Title

And distinct??

31. Zarkon Group Title

yes

32. IsTim Group Title

Awesome. Thank you. This was relatively simple after all...


text:
# 65 (number): Wikis

Note: Many of our articles have direct quotes from sources you can cite, within the Wikipedia article! This article doesn't yet, but we're working on it! See more info or our list of citable articles.

# Encyclopedia

Updated live from Wikipedia, last check: May 23, 2013 10:51 UTC (37 seconds ago)

65
Cardinal sixty-five
Ordinal 65th
(sixty-fifth)
Factorization $5 \cdot 13$
Divisors 1, 5, 13, 65
Roman numeral LXV
Binary 10000012
Octal 1018
Duodecimal 5512

65 (sixty-five) is the natural number following 64 and preceding 66.

## In mathematics

Sixty-five is the 19th semiprime and the 3rd of the form (5.q)it is an octagonal number. It is also a Cullen number. Given 65, the Mertens function returns 0.

Sixty-five has an aliquot sum of 19 and is the first composite number in the 19-aliquot tree.

This number is the magic constant of 5 by 5 normal magic square:

$\begin{bmatrix} 17 & 24 & 1 & 8 & 15 \ 23 & 5 & 7 & 14 & 16 \ 4 & 6 & 13 & 20 & 22 \ 10 & 12 & 19 & 21 & 3 \ 11 & 18 & 25 & 2 & 9 \end{bmatrix}.$

This number is also the magic constant of n-Queens Problem for n = 5.

65 is the smallest integer that can be expressed as a sum of two distinct positive squares in two ways, 65 = 82 + 12 = 72 + 42.

It appears in the Padovan sequence, preceded by the terms 28, 37, 49 (it is the sum of the first two of these).

There are only 65 known Euler's idoneal numbers.

65=15+24+33+42+51

## In science

### Astronomy

• The Saros number of the solar eclipse series which began on -749 April 24 and ended on 513 May. The duration of Saros series 65 was 1262.1 years, and it contained 71 solar eclipses.
• The Saros number of the lunar eclipse series which began on -736 August 11 and ended on 797 February. The duration of Saros series 65 was 1532.5 years, and it contained 86 lunar eclipses.

Sixty-five is:

## Historical years

65 A.D., 65 B.C., 1965, 2065, etc.

# Simple English

Sixty-five is a number. It comes between sixty-four and sixty-six, and is an odd number. It is divisible by 5, and 13.

## 65 in some places

• +65 is the International Calling Code for Singapore


text:
# 22.7 Equivalent k-HD-TA for a given 1-NTA

Question: Given a non-deterministic timed automata with 1 clock(1-NTA) and reachability acceptance condition, decide whether there exists an equivalent history deterministic timed automata with k clocks ($k$-HDTA), where $$k$$ can be fixed or arbitrary?

Motivation: The classes of 1-NTA, deterministic timed automata (DTA), history-deterministic timed automata (HDTA) all have decidable language inclusion problems, though the complexity of inclusion for 1-NTA is non-primitive recursive. However, the class 1-NTA is incomparable to DTA and HDTA.

Whether a given 1-NTA is equivalent to some $k$-DTA is decidable, but it the problem becomes undecidable if the number of clocks in the DTA is not fixed beforehand. The proposed question is simply extending this to HD case instead of deterministic.

Remarks: It is unknown whether HDTA is a strictly bigger class than DTA. Showing that the two classes (DTA and HDTA) are expressively equivalent would settle the problem by reduction to the DTA case.


text:
# What colour change would NaCl(aq) cause to added litmus paper?

Litmus paper may change colour depending on the pH of the solution in which it is placed. We has $N a C l \left(a q\right)$, which as the salt of a strong acid and strong base does not affect aqueous $p H$,
On the other hand, if we examined the $p H$ of say $N a F \left(a q\right)$, we would find that the solution was SLIGHTLY basic due to the hydrolytic action of the fluoride ion:
${F}^{-} + {H}_{2} O \left(l\right) r i g h t \le f t h a r p \infty n s H F \left(a q\right) + H {O}^{-}$


text:
# American Institute of Mathematical Sciences

• Previous Article
Almost-periodic perturbations of non-hyperbolic equilibrium points via Pöschel-Rüssmann KAM method
• CPAA Home
• This Issue
• Next Article
On large potential perturbations of the Schrödinger, wave and Klein–Gordon equations
January  2020, 19(1): 587-607. doi: 10.3934/cpaa.2020028

## Potential well and multiplicity of solutions for nonlinear Dirac equations

 1 School of Mathematical Sciences, Beijing Normal University, Laboratory of Mathematics and Complex System, Ministry of Education, 100875 Beijing, China 2 Institute of Mathematics, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, 100190 Beijing, China 3 Center for Applied Mathematics, Tianjin University, 300072 Tianjin, China

Received  September 2018 Revised  September 2018 Published  July 2019

In this paper we consider the semi-classical solutions of a massive Dirac equations in presence of a critical growth nonlinearity
 $-i\hbar \sum\limits_{k = 1}^{3}\alpha_k\partial_k w+a\beta w+V(x)w = f(|w|)w.$
Under a local condition imposed on the potential
 $V$
, we relate the number of solutions with the topology of the set where the potential attains its minimum. In the proofs we apply variational methods, penalization techniques and Ljusternik-Schnirelmann theory.
Citation: Yu Chen, Yanheng Ding, Tian Xu. Potential well and multiplicity of solutions for nonlinear Dirac equations. Communications on Pure & Applied Analysis, 2020, 19 (1) : 587-607. doi: 10.3934/cpaa.2020028
##### References:

show all references

##### References:
 [1] Claude Bardos, Nicolas Besse. The Cauchy problem for the Vlasov-Dirac-Benney equation and related issues in fluid mechanics and semi-classical limits. Kinetic & Related Models, 2013, 6 (4) : 893-917. doi: 10.3934/krm.2013.6.893 [2] Yuanhong Wei, Yong Li, Xue Yang. On concentration of semi-classical solitary waves for a generalized Kadomtsev-Petviashvili equation. Discrete & Continuous Dynamical Systems - S, 2017, 10 (5) : 1095-1106. doi: 10.3934/dcdss.2017059 [3] Lihui Chai, Shi Jin, Qin Li. Semi-classical models for the Schrödinger equation with periodic potentials and band crossings. Kinetic & Related Models, 2013, 6 (3) : 505-532. doi: 10.3934/krm.2013.6.505 [4] Xu Zhang. On the concentration of semiclassical states for nonlinear Dirac equations. Discrete & Continuous Dynamical Systems - A, 2018, 38 (11) : 5389-5413. doi: 10.3934/dcds.2018238 [5] Jean Dolbeault, Robert Stańczy. Bifurcation diagrams and multiplicity for nonlocal elliptic equations modeling gravitating systems based on Fermi--Dirac statistics. Discrete & Continuous Dynamical Systems - A, 2015, 35 (1) : 139-154. doi: 10.3934/dcds.2015.35.139 [6] Wen Feng, Milena Stanislavova, Atanas Stefanov. On the spectral stability of ground states of semi-linear Schrödinger and Klein-Gordon equations with fractional dispersion. Communications on Pure & Applied Analysis, 2018, 17 (4) : 1371-1385. doi: 10.3934/cpaa.2018067 [7] Maria J. Esteban, Eric Séré. An overview on linear and nonlinear Dirac equations. Discrete & Continuous Dynamical Systems - A, 2002, 8 (2) : 381-397. doi: 10.3934/dcds.2002.8.381 [8] Marek Galewski, Shapour Heidarkhani, Amjad Salari. Multiplicity results for discrete anisotropic equations. Discrete & Continuous Dynamical Systems - B, 2018, 23 (1) : 203-218. doi: 10.3934/dcdsb.2018014 [9] Xiaoyan Lin, Xianhua Tang. Solutions of nonlinear periodic Dirac equations with periodic potentials. Discrete & Continuous Dynamical Systems - S, 2019, 12 (7) : 2051-2061. doi: 10.3934/dcdss.2019132 [10] Giuseppe Maria Coclite, Helge Holden. Ground states of the Schrödinger-Maxwell system with dirac mass: Existence and asymptotics. Discrete & Continuous Dynamical Systems - A, 2010, 27 (1) : 117-132. doi: 10.3934/dcds.2010.27.117 [11] Minbo Yang, Yanheng Ding. Existence and multiplicity of semiclassical states for a quasilinear Schrödinger equation in $\mathbb{R}^N$. Communications on Pure & Applied Analysis, 2013, 12 (1) : 429-449. doi: 10.3934/cpaa.2013.12.429 [12] Norimichi Hirano, Wen Se Kim. Multiplicity and stability result for semilinear parabolic equations. Discrete & Continuous Dynamical Systems - A, 1996, 2 (2) : 271-280. doi: 10.3934/dcds.1996.2.271 [13] Nikolaos Bournaveas, Timothy Candy, Shuji Machihara. A note on the Chern-Simons-Dirac equations in the Coulomb gauge. Discrete & Continuous Dynamical Systems - A, 2014, 34 (7) : 2693-2701. doi: 10.3934/dcds.2014.34.2693 [14] Noboru Okazawa, Kentarou Yoshii. Linear evolution equations with strongly measurable families and application to the Dirac equation. Discrete & Continuous Dynamical Systems - S, 2011, 4 (3) : 723-744. doi: 10.3934/dcdss.2011.4.723 [15] Hyungjin Huh. Self-similar solutions to nonlinear Dirac equations and an application to nonuniqueness. Evolution Equations & Control Theory, 2018, 7 (1) : 53-60. doi: 10.3934/eect.2018003 [16] Changhun Yang. Scattering results for Dirac Hartree-type equations with small initial data. Communications on Pure & Applied Analysis, 2019, 18 (4) : 1711-1734. doi: 10.3934/cpaa.2019081 [17] Hui Zhang, Jun Wang, Fubao Zhang. Semiclassical states for fractional Choquard equations with critical growth. Communications on Pure & Applied Analysis, 2019, 18 (1) : 519-538. doi: 10.3934/cpaa.2019026 [18] Yonggeun Cho, Tohru Ozawa. On radial solutions of semi-relativistic Hartree equations. Discrete & Continuous Dynamical Systems - S, 2008, 1 (1) : 71-82. doi: 10.3934/dcdss.2008.1.71 [19] Yi-hsin Cheng, Tsung-Fang Wu. Multiplicity and concentration of positive solutions for semilinear elliptic equations with steep potential. Communications on Pure & Applied Analysis, 2016, 15 (6) : 2457-2473. doi: 10.3934/cpaa.2016044 [20] Peng Chen, Xiaochun Liu. Multiplicity of solutions to Kirchhoff type equations with critical Sobolev exponent. Communications on Pure & Applied Analysis, 2018, 17 (1) : 113-125. doi: 10.3934/cpaa.2018007

2018 Impact Factor: 0.925


text:
# Binomial distribution problem

Tags:
1. Apr 12, 2016

### MarcusAu314

• Thread moved from a different forums, so no HH Template is shown
Right now I'm having a problem with a statistics problem. More specifically with a binomial distribution problem.

The problem says:
There is a family composed by 8 children. Calculate the probability that 3 of them are girls

As far as I know, binomial distribution formula says:
$p(x=k)=\binom{n}{k}(p)^{k}(1-p)^{n-k}$

In which:
*n is the number of trials
*k is the number of success
*p is the probability of success
*(1-p) is the probability of failure, sometimes denoted as q in some textbooks

I know two things, since we are talking about eight children then n=8. The number of success is k=3, therefore the number of failures must be n-k= 5. At this point I feel I'm going well. However the problem begins that problem isn't giving me the probability of success (p). I first tried to calculate it by knowing that if 8 children means 100% of all the trials then 3 girls means 37.5% (I found this by rule of three); yet I'm not completely sure if that's the right way of finding the probability of success p when problem isn't giving it to us.

Thanks.

2. Apr 12, 2016

### Ray Vickson

I think they want you to assume that boy and girl babies are equally likely, so that $p = q = 1/2$. In reality that is not quite true; if you want, you can look up the boy/girl birth rates in a database, and use that instead. However, assuming $p = q = 1/2$ in such problems is pretty standard in an introductory course.

Last edited: Apr 13, 2016
3. Apr 12, 2016

### haruspex

Exactly three in total, at least three in total, or a specific three? Ok, I agree they probably mean exactly three in total.


text:
# Divided Prime

## 输入输出样例

### 输入样例 #1

2
3 2
5 7 7
5 7
4 2
5 7 7 7
5 7

### 输出样例 #1

YES
NO

## 说明

$1 \le N \le 100000$ $0 \le M \le N$ $1 \le a_i,b_i \le 10^{12}$ $1 \le T \le 10$ $\sum N \le 100000$


text: dc.creator Sakurai, K. en_US dc.creator Taieb, G. en_US dc.creator Broida, H. P. en_US dc.date.accessioned 2006-06-15T13:44:24Z dc.date.available 2006-06-15T13:44:24Z dc.date.issued 1976 en_US dc.identifier 1976-MM-10 en_US dc.identifier.uri http://hdl.handle.net/1811/9601 dc.description Author Institution: Department of Physics and quantum Institute, University of California en_US dc.description.abstract Calcium diatomic molecules were produced in a heat-pipe oven at temperatures from $1000-1200^\circ K$ . Photoluminescence of $Ca_{2}$ has been studied with various laser sources and with white light. Discrete, sharp line spectra are attributed to known bound states of $Ca_{2}$. A broad continuum emission with an oscillatory intensity distribution is associated with emission to the repulsive part of the lower state. A weak continuum emission to another repulsive state also has been observed. The observed strong emission from the atomic resonance line at 422.7 and 657.3 nm probably is caused by dissociation of excited $Ca_{2}$ molecules. The laser-excited photoluminescence experiments suggest that the ground state of $Ca_{2}$ is repulsive and that the lowest observed bound state of $Ca_{2}$ correlates with the $^{3}p$ atomic state. en_US dc.format.extent 113481 bytes dc.format.mimetype image/jpeg dc.language.iso English en_US dc.publisher Ohio State University en_US dc.title LASER PHOTOLUMINESCENCE OF CALCIUM MOLECULES en_US dc.type article en_US


text:
# Source code for gemseo.mlearning.core.calibration

# Copyright 2021 IRT Saint Exupéry, https://www.irt-saintexupery.com
#
# This program is free software; you can redistribute it and/or
# modify it under the terms of the GNU Lesser General Public
# License version 3 as published by the Free Software Foundation.
#
# This program is distributed in the hope that it will be useful,
# but WITHOUT ANY WARRANTY; without even the implied warranty of
# MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the GNU
# Lesser General Public License for more details.
#
# You should have received a copy of the GNU Lesser General Public License
# along with this program; if not, write to the Free Software Foundation,
# Inc., 51 Franklin Street, Fifth Floor, Boston, MA  02110-1301, USA.
# Contributors:
#    INITIAL AUTHORS - API and implementation and/or documentation
#        :author: Matthias De Lozzo
#    OTHER AUTHORS   - MACROSCOPIC CHANGES
"""Calibration of a machine learning algorithm.

A machine learning algorithm depends on hyper-parameters,
e.g. the number of clusters for a clustering algorithm,
the regularization constant for a regression model,
the kernel for a Gaussian process regression, ...
Its ability to generalize the information learned during the training stage,
and thus to avoid over-fitting,
which is an over-reliance on the learning data set,
depends on the values of these hyper-parameters.
Thus,
the hyper-parameters minimizing the learning quality measure are rarely
those minimizing the generalization one.
Classically,
the generalization one decreases before growing again as the model becomes more complex,
while the learning error keeps decreasing.
This phenomenon is called the curse of dimensionality.

In this module,
the :class:.MLAlgoCalibration class aims to calibrate the hyper-parameters
in order to minimize this measure of the generalization quality
over a calibration parameter space.
This class relies on the :class:.MLAlgoAssessor class
which is a discipline (:class:.MDODiscipline)
built from a machine learning algorithm (:class:.MLAlgo),
a dataset (:class:.Dataset),
a quality measure (:class:.MLQualityMeasure)
and various options for the data scaling,
the quality measure
and the machine learning algorithm.
The inputs of this discipline are hyper-parameters of the machine learning algorithm
while the output is the quality criterion.
"""
from __future__ import annotations

from typing import Dict
from typing import Iterable
from typing import Union

from numpy import argmin
from numpy import array
from numpy import ndarray

from gemseo.algos.design_space import DesignSpace
from gemseo.algos.doe.doe_factory import DOEFactory
from gemseo.core.dataset import Dataset
from gemseo.core.discipline import MDODiscipline
from gemseo.core.doe_scenario import DOEScenario
from gemseo.core.mdo_scenario import MDOScenario
from gemseo.core.scenario import Scenario
from gemseo.core.scenario import ScenarioInputDataType
from gemseo.mlearning.core.factory import MLAlgoFactory
from gemseo.mlearning.core.ml_algo import MLAlgo
from gemseo.mlearning.core.ml_algo import MLAlgoParameterType
from gemseo.mlearning.core.ml_algo import TransformerType
from gemseo.mlearning.qual_measure.quality_measure import MLQualityMeasure

MeasureOptionsType = Dict[str, Union[bool, int, Dataset]]

[docs]class MLAlgoAssessor(MDODiscipline):
"""Discipline assessing the quality of a machine learning algorithm.

This quality depends on the values of parameters to calibrate with the
:class:.MLAlgoCalibration.
"""

algo: str
"""The name of a machine learning algorithm."""

measure: MLQualityMeasure
"""The measure to assess the machine learning algorithm."""

measure_options: dict[str, int | Dataset]
"""The options of the quality measure."""

parameters: list[str]
"""The parameters of the machine learning algorithm."""

dataset: Dataset
"""The learning dataset."""

transformer: TransformerType
"""The transformation strategy for data groups."""

algos: list[MLAlgo]
"""The instances of the machine learning algorithm (one per execution of the machine
learning algorithm assessor)."""

CRITERION = "criterion"
LEARNING = "learning"
MULTIOUTPUT = "multioutput"

def __init__(
self,
algo: str,
dataset: Dataset,
parameters: Iterable[str],
measure: type[MLQualityMeasure],
measure_options: MeasureOptionsType | None = None,
transformer: TransformerType = MLAlgo.IDENTITY,
**algo_options: MLAlgoParameterType,
) -> None:
"""
Args:
algo: The name of a machine learning algorithm.
dataset: A learning dataset.
parameters: The parameters of the machine learning algorithm to calibrate.
measure: A measure to assess the machine learning algorithm.
measure_options: The options of the quality measure.
If "multioutput" is missing,
it is added with False as value.
If None, do not use quality measure options.
transformer: The strategies
to transform the variables.
The values are instances of :class:.Transformer
while the keys are the names of
either the variables
or the groups of variables,
e.g. "inputs" or "outputs"
in the case of the regression algorithms.
If a group is specified,
the :class:.Transformer will be applied
to all the variables of this group.
If :attr:~.MLAlgo.IDENTITY, do not transform the variables.

**algo_options: The options of the machine learning algorithm.

Raises:
ValueError: If the measure option "multioutput" is True.
"""
super().__init__()
self.input_grammar.update(parameters)
self.output_grammar.update([self.CRITERION, self.LEARNING])
self.algo = algo
self.measure = measure
self.measure_options = measure_options or {}
self.parameters = algo_options
self.data = dataset
self.transformer = transformer
self.algos = []

if self.measure_options.get("multioutput", False):
raise ValueError("MLAlgoAssessor does not support multioutput.")
self.measure_options[self.MULTIOUTPUT] = False

def _run(self) -> None:
"""Run method.

This method creates a new instance of the machine learning algorithm, from the
hyper-parameters stored in the local_data attribute of the
:class:.MLAlgoAssessor. It trains it on the learning dataset and measures its
quality with the :class:.MLQualityMeasure.
"""
inputs = self.get_input_data()
for index in inputs:
if len(inputs[index]) == 1:
inputs[index] = inputs[index][0]
self.parameters.update(inputs)
factory = MLAlgoFactory()
algo = factory.create(
self.algo, data=self.data, transformer=self.transformer, **self.parameters
)
algo.learn()
measure = self.measure(algo)
learning = measure.evaluate(multioutput=False)
criterion = measure.evaluate(**self.measure_options)
self.store_local_data(criterion=array([criterion]), learning=array([learning]))
self.algos.append(algo)

[docs]class MLAlgoCalibration:
"""Calibration of a machine learning algorithm."""

algo_assessor: MLAlgoAssessor
"""The assessor for the machine learning algorithm."""

calibration_space: DesignSpace
"""The space defining the calibration variables."""

maximize_objective: bool
"""Whether to maximize the quality measure."""

dataset: Dataset
"""The learning dataset."""

optimal_parameters: dict[str, ndarray]
"""The optimal parameters for the machine learning algorithm."""

optimal_criterion: float
"""The optimal quality measure."""

optimal_algorithm: MLAlgo
"""The optimal machine learning algorithm."""

scenario: Scenario
"""The scenario used to calibrate the machine learning algorithm."""

def __init__(
self,
algo: str,
dataset: Dataset,
parameters: Iterable[str],
calibration_space: DesignSpace,
measure: MLQualityMeasure,
measure_options: MeasureOptionsType | None = None,
transformer: TransformerType = MLAlgo.IDENTITY,
**algo_options: MLAlgoParameterType,
) -> None:
"""
Args:
algo: The name of a machine learning algorithm.
dataset: A learning dataset.
parameters: The parameters of the machine learning algorithm
to calibrate.
calibration_space: The space defining the calibration variables.
measure: A measure to assess the machine learning algorithm.
measure_options: The options of the quality measure.
If None, do not use the quality measure options.
transformer: The strategies
to transform the variables.
The values are instances of :class:.Transformer
while the keys are the names of
either the variables
or the groups of variables,
e.g. "inputs" or "outputs"
in the case of the regression algorithms.
If a group is specified,
the :class:.Transformer will be applied
to all the variables of this group.
If :attr:~.MLAlgo.IDENTITY, do not transform the variables.
**algo_options: The options of the machine learning algorithm.
"""
disc = MLAlgoAssessor(
algo,
dataset,
parameters,
measure,
measure_options,
transformer,
**algo_options,
)
self.algo_assessor = disc
self.calibration_space = calibration_space
self.maximize_objective = not measure.SMALLER_IS_BETTER
self.dataset = None
self.optimal_parameters = None
self.optimal_criterion = None
self.optimal_algorithm = None
self.scenario = None

[docs]    def execute(
self,
input_data: ScenarioInputDataType,
) -> None:
"""Calibrate the machine learning algorithm from a driver.

The driver can be either a DOE or an optimizer.

Args:
input_data: The driver properties.
"""
doe_factory = DOEFactory()

if doe_factory.is_available(input_data["algo"]):
self.scenario = DOEScenario(
[self.algo_assessor],
"DisciplinaryOpt",
self.algo_assessor.CRITERION,
self.calibration_space,
maximize_objective=self.maximize_objective,
)
else:
self.scenario = MDOScenario(
[self.algo_assessor],
"DisciplinaryOpt",
self.algo_assessor.CRITERION,
self.calibration_space,
maximize_objective=self.maximize_objective,
)
self.scenario.execute(input_data)
x_opt = self.scenario.design_space.get_current_value(as_dict=True)
f_opt = self.scenario.get_optimum().f_opt
self.dataset = self.scenario.export_to_dataset(by_group=False, opt_naming=False)
algo_opt = self.algos[argmin(self.get_history(self.algo_assessor.CRITERION))]
self.optimal_parameters = x_opt
self.optimal_criterion = f_opt
self.optimal_algorithm = algo_opt

[docs]    def get_history(
self,
name: str,
) -> ndarray:
"""Return the history of a variable.

Args:
name: The name of the variable.

Returns:
The history of the variable.
"""
if self.dataset is not None:
if name == self.algo_assessor.CRITERION and self.maximize_objective:
return -self.dataset.data["-" + name]
else:
return self.dataset.data[name]

@property
def algos(self) -> MLAlgo:
"""The trained machine learning algorithms."""
return self.scenario.disciplines[0].algos


text:
# Is there preferential attachment in the growth of early semantic noun networks?

Hills, T. and Maoune, J. and Sheya, A. and Maoune, M. and Smith, L.. (2008) Is there preferential attachment in the growth of early semantic noun networks? In: Proceedings of the 30th Annual Conference of the Cognitive Science Society. Austin, pp. 155-160.

Full text not available from this repository.

Official URL: http://edoc.unibas.ch/dok/A5265640


text:
Not a member?

# Discussion Groups | Comp.DSP | Wavelets...

There are 7 messages in this thread.

You are currently looking at messages 1 to .

Is this discussion worth a thumbs up?

0

# Wavelets... - brettcooper - 2012-06-27 12:16:00

Hello Forum,

I have just started studying wavelets and wavelet transform.
While the STFT breaks the signal down into equal time duration segments,
the wavelet transform uses wavelets that when they have high frequency
(compressed) give great time resolution, i.e. short time window.
But a wavelet with a low frequency implies a stretched up wavelet and a
long time window...

It seems that wavelets can do great frequency and time resolution at high
frequencies and modest time resolution at lower frequencies...
Is that correct?

Wavelets are orthogonal to each other. could you remind me the benefit of
decomposing a signal in orthogonal basis functions aside from the fact that
each basis function gives a zero inner product with another basis function
of different frequency?
the purpose of all transforms is to find the complex coefficients to give
the basis functions for the best reconstruction of the signal...

In Fourier theory the basis functions have complex coefficients containing
amplitude and phase....do wavelets have phase information? Or just the
translation coefficient?
thanks,
Brett

______________________________

# Re: Wavelets... - Thomas Richter - 2012-06-27 13:32:00

Am 27.06.2012 18:16, schrieb brettcooper:

> It seems that wavelets can do great frequency and time resolution at high
> frequencies and modest time resolution at lower frequencies...
> Is that correct?

The "uncertainty" relation still holds true for wavelets, so you may ask
about the support of the wavelet functions in time or frequency domain.
Compared to a Fourier basis - where the basis functions have all the
same support in the frequency domain, wavelets have a relatively small
domain for lower frequencies (thus better resolution) and a larger
domain (lower resolution) for higher frequencies. This implies that time
resolution is just the inverse of that.

> Wavelets are orthogonal to each other.

No, not necessarily. Wavelets can be both orthogonal and non-orthogonal.
Actually, the latter are the more interesting since non-orthgonal
wavelets may have more interesting symmetries than orthogonal ones.

> could you remind me the benefit of
> decomposing a signal in orthogonal basis functions aside from the fact that
> each basis function gives a zero inner product with another basis function
> of different frequency?

That is an interesting property because it implies that you can
implement a wavelet analysis by means of an inner product with the
synthesis functions. For non-orthogonal filters, the relation between
analysis and synthesis filters is not so simple.

> In Fourier theory the basis functions have complex coefficients containing
> amplitude and phase....do wavelets have phase information? Or just the
> translation coefficient?

I don't quite understand how one could define "phase" without having a
planar wave and thus a Fourier decomposition, thus I don't quite see how
this concept carries over directly. Still, if you have complex signals,
you could define the "phase" in the same way, namely by representing the
wavelet coefficients (from the wavelet analysis) as r * exp(i\phi),
where \phi would be the phase. But it is not exactly the same type of
"phase shift = transition in time domain" as you know it for Fourier.

Greetings,
Thomas

______________________________

# Re: Wavelets... - brettcooper - 2012-06-28 10:14:00

Thanks Thomas.

I just read this online: "....Short Term Fourier Transform (STFT) -
provides both time and frequency information, but resolves all frequencies
equally...."

I get that.

"...Wavelet transform - provides good time resolution and poor frequency
resolution at high frequencies and good frequency resolution and poor time
resolution at low frequencies. Useful approach when signal at hand has high
frequency components for short duration and low frequency components for
long duration as in ECG....."

So a compressed, high frequency wavelets automatically have a shorter
duration, hence a good time resolution... but why do they have poor
frequency resolution?  Vice versa for the low frequency wavelets...

Poor frequency resolution means that we need wavelets of more frequencies
to suitably describe a certain event in the signal. Is it because short
duration events are usually more complicated in shape than long duration
events (smooth changes)?

for example, a chirp signal with increasing frequency: I feel that the
wavelet transform should work equally well at low and high frequencies...

Also, in communication theory, a carrier is FM modulated and demodulated to
get the instantaneous frequency (Which is related to the transmitted
message). Do they use wavelets to extract the instantaneous frequency out
of the modulated signal? If not, how do they do it?

Brett

______________________________

# Re: Wavelets... - maury - 2012-06-28 10:37:00

On Thursday, June 28, 2012 9:14:01 AM UTC-5, brettcooper wrote:
> Thanks Thomas.
>
> I just read this online: "....Short Term Fourier Transform (STFT) -
> provides both time and frequency information, but resolves all frequencies
> equally...."
>
> I get that.
>
> "...Wavelet transform - provides good time resolution and poor frequency
> resolution at high frequencies and good frequency resolution and poor time
> resolution at low frequencies. Useful approach when signal at hand has high
> frequency components for short duration and low frequency components for
> long duration as in ECG....."
>
> So a compressed, high frequency wavelets automatically have a shorter
> duration, hence a good time resolution... but why do they have poor
> frequency resolution?  Vice versa for the low frequency wavelets...
>
> Poor frequency resolution means that we need wavelets of more frequencies
> to suitably describe a certain event in the signal. Is it because short
> duration events are usually more complicated in shape than long duration
> events (smooth changes)?
>
> for example, a chirp signal with increasing frequency: I feel that the
> wavelet transform should work equally well at low and high frequencies...
>
> Also, in communication theory, a carrier is FM modulated and demodulated to
> get the instantaneous frequency (Which is related to the transmitted
> message). Do they use wavelets to extract the instantaneous frequency out
> of the modulated signal? If not, how do they do it?
>
> Brett

Go to Google.com type in TIME FREQUENCY UNCERTAINITY PRINCIPAL

______________________________

# Re: Wavelets... - Thomas Richter - 2012-06-28 13:45:00

Am 28.06.2012 16:14, schrieb brettcooper:
> Thanks Thomas.

> "...Wavelet transform - provides good time resolution and poor frequency
> resolution at high frequencies and good frequency resolution and poor time
> resolution at low frequencies. Useful approach when signal at hand has high
> frequency components for short duration and low frequency components for
> long duration as in ECG....."
>
> So a compressed, high frequency wavelets automatically have a shorter
> duration, hence a good time resolution... but why do they have poor
> frequency resolution?  Vice versa for the low frequency wavelets...

Because you have a fundamental inequality between time resolution and
frequency resolution that forbids you to have both at the same time.
Actually, it is the same inequality in physics where the "frequency
space" is also the "momentum space" of quantum particles, and the "time
space" is the "position space" of the particle, where the Fourier
transform transforms between the two representations. It is there (in
physics) called the "uncertainty relation" because it implies that you
cannot measure both the position *and* the momentum of a quantum
particle sharply.

The same relation holds in signal processing as well, simply because the
mathematical transformation is the same:

To define a frequency sharply, you need an infinitely extended "planar
wave", i.e. f(x) = exp(i k x), which has a fixed 'k', but is obviously
extended all over the space. Vice versa, if you want a sharp position in
the time domain, the delta "function" (actually, distribution) does
that. Its image in the Fourier domain, after Fourier transformation, is
of course the plane wave again, \hat{f}(k) = exp(-i k x) where x is now
the parameter where the "peak" is and k the argument of the function
(the other way around than above). It is hence infinitely extended in
'k' (frequency, momentum) space, but located in the time domain.

Wavelets are compromises between the two extreme, neither localized in
time nor in the frequency domain.

> Poor frequency resolution means that we need wavelets of more frequencies
> to suitably describe a certain event in the signal.

How many basis functions you need is entirely up to the signal you want
to describe. One cannot say, in general, you need "more" or "less". It
is the nature of the signal that tells you. If the signal is a slightly
distorted plane wave, a Fourier representation would be more suitable.

> Is it because short
> duration events are usually more complicated in shape than long duration
> events (smooth changes)?

Signals of short duration have a more complex spectrum, i.e. have a less
sharp frequency distribution. Shorter signals = "more overtones", and
"purer signals" require longer samplings, i.e. are not localized in time
domain.

> for example, a chirp signal with increasing frequency: I feel that the
> wavelet transform should work equally well at low and high frequencies...

That cannot be answered in general. For such a signal, i.e. a slow
variation in the time domain, I would believe that a Fourier description
is more adequate. I don't think one can compute the spectrum of such a
signal explicitly, but I haven't tried either. An approximation would be
feasible.

> Also, in communication theory, a carrier is FM modulated and demodulated to
> get the instantaneous frequency (Which is related to the transmitted
> message). Do they use wavelets to extract the instantaneous frequency out
> of the modulated signal? If not, how do they do it?

Sorry, I don't think I got this question. Wavelets are not a tool for FM
modulation, at least I wouldn't know how to apply them to this problem
without thinking more carefully about it. Wavelets are, just to give one
example, a good model to describe natural images. JPEG 2000 uses for
example a wavelet decomposition to describe the image data. It is here a
good model since the frequency resolution of wavelets gets lower for
high frequencies and higher at low frequencies, which is very much how
the eye-brain visual system behaves. So it is a reasonably good model of
how we "see" the world.

So long,
Thomas

______________________________

# Re: Wavelets... - Alexander Petrov - 2012-06-29 03:47:00

>Thanks Thomas.
>
>I just read this online: "....Short Term Fourier Transform (STFT) -
>provides both time and frequency information, but resolves all
frequencies
>equally...."
>
>I get that.
>
>"...Wavelet transform - provides good time resolution and poor frequency
>resolution at high frequencies and good frequency resolution and poor
time
>resolution at low frequencies. Useful approach when signal at hand has
high
>frequency components for short duration and low frequency components for
>long duration as in ECG....."
>
>So a compressed, high frequency wavelets automatically have a shorter
>duration, hence a good time resolution... but why do they have poor
>frequency resolution?  Vice versa for the low frequency wavelets...
>
>Poor frequency resolution means that we need wavelets of more frequencies
>to suitably describe a certain event in the signal. Is it because short
>duration events are usually more complicated in shape than long duration
>events (smooth changes)?
>
>for example, a chirp signal with increasing frequency: I feel that the
>wavelet transform should work equally well at low and high frequencies...
>
>Also, in communication theory, a carrier is FM modulated and demodulated
to
>get the instantaneous frequency (Which is related to the transmitted
>message). Do they use wavelets to extract the instantaneous frequency out
>of the modulated signal? If not, how do they do it?
>
>Brett
>

http://en.wikipedia.org/wiki/Wavelet_packet_decomposition

______________________________

# Re: Wavelets... - brettcooper - 2012-06-29 10:16:00

Thank you again....

i guess I got confused with the fact that a FM modulated signal, with the
concept of instantaneous frequency, can have a specific single frequency at
every instant in time (that is its instantaneous freq.)
But the Fourier transform of a FM modulated signal clearly shows a wide
bandwidth....

The concept of frequency has multiple definitions. for instance, a pure
sinusoid with a Gaussian envelope forms a finite time durations signal with
a nonzero frequency bandwidth...
But the interzero distance in this signal is constant, i.e. the
instantaneous frequency is one and constant while the Fourier frequency is
not just one but a whole bandwidth...

Brett

______________________________


text:
# Question #931d4

Apr 30, 2017

We have to use that the geometric series is of the form:

$a + a r + a {r}^{2} + a {r}^{3} + a {r}^{4} + \ldots$

and that we know that the whole sum is $13.5$ and the sum of the first three terms $a + a r + a {r}^{2} = 13$

#### Explanation:

So we have:

$a + a r + a {r}^{2} + a {r}^{3} + a {r}^{4} + \ldots = 13.5$

and

$a + a r + a {r}^{2} = 13$

Subtracting we have:

$\left(a + a r + a {r}^{2} + a {r}^{3} + a {r}^{4} + \ldots\right) - \left(a + a r + a {r}^{2}\right) = 0.5$, that is:

$a {r}^{3} + a {r}^{4} + a {r}^{5} + a {r}^{6} + a {r}^{7} + \ldots = 0.5$

Hence, taking $r$ as a common factor:

${r}^{3} \cdot \left(a + a r + a {r}^{2} + a {r}^{3} + a {r}^{4} + \ldots\right) = 0.5$

But we know that that the expression between brackets:

$a + a r + a {r}^{2} + a {r}^{3} + a {r}^{4} + \ldots = 13.5$

so we have:

${r}^{3} \cdot 13.5 = 0.5$, that is:

${r}^{3} = \frac{0.5}{13.5}$, that is:

${r}^{3} = \frac{\frac{1}{2}}{\frac{27}{2}} = \frac{1}{27}$, and then

$r = \frac{1}{3}$

But we know that $a + a r + a {r}^{2} = 13$, so:

$a + a \frac{1}{3} + a \frac{1}{9} = 13$, and that means:

$\frac{9 a + 3 a + a}{9} = 13$, so:

$\frac{13 a}{9} = 13$, so $\frac{a}{9} = 1$, and from that we get:

$a = 9$, which is the first term