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# Flux flow spin Hall effect in type-II superconductors with spin-splitting field

## Abstract

We predict the very large spin Hall effect in type-II superconductors whose mechanism is drastically different from the previously known ones. We find that in the flux-flow regime the spin is transported by the spin-polarized Abrikosov vortices moving under the action of the Lorenz force in the direction perpendicular to the applied electric current. Due to the large vortex velocities the spin Hall angle can be of the order of unity in realistic systems based on the high-field superconductors, superconductor/ferromagnet hybrid structures or the recently developed superconductor/ferromagnetic insulator proximity structures. We propose the realization of high-frequency pure spin current generator based on the periodic structure of moving vortex lattices. We find the patterns of charge imbalance and spin accumulation generated by moving vortices, which can be used for the electrical detection of individual vortex motion. The new mechanism of inverse flux-flow spin Hall effect is found based on the driving force acting on the vortices in the presence of injected spin current which results in the generation of transverse voltage.

## Introduction

The spin Hall effect (SHE) is currently one of the basic tools in spintronics used for the generation and detection of pure spin currents1. Although it has quite a rich variety of applications, from the fundamental point of view there have been only two known mechanisms leading to the spin Hall effect: (i) the spin-orbital interaction in semiconductors and heavy metals and (ii) the Zeeman spin splitting in graphene close to the neutrality point making the electrons and holes to carry different spin polarizations2,3,4. Here we suggest the third fundamental mechanism combining the specific properties of the electronic spectrum in superconductors with spin-splitting field and the coherent dynamics of the superconducting order parameter manifested through the flux flow of Abrikosov vortices under the action of the external transport current.

The non-equilibrium properties of superconductors with spin-splitting fields have become a hot topic in the field of superconductivity5. Such systems are characterized by the spin-dependent electron-hole asymmetry of Bogolubov quasiparticles6. Recently it has been realized that this feature allows for the generation of long-range spin accumulation5,7,8,9,10,11, which is robust against the usual spin-flip and spin-orbital scattering relaxations. This mechanism explains many experimental observations of long-range non-local spin signals in mesoscopic superconducting wires generated by the injected current from the ferromagnetic or even non-ferromagnetic electrodes12,13,14,15. In this paper we demonstrate the possibility of not only the long-range spin accumulation but also the non-decaying pure spin current generation using the properties of superconductors with spin-splitting fields.

In principle, the paramagnetic spin-splitting of Bogolubov quasiparticles appears inevitably due to the Zeeman effect in any superconductor subject to the magnetic field12,14,16,17. However, the magnetic field simultaneously leads to the orbital effect, inducing the center-of mass motion of the Cooper pairs due to the Meissner effect. The relative magnitude of the paramagnetic shift and the orbital kinetic energy of the Cooper pair is determined by the parameter introduced by Maki16 (referred later as the Maki parameter) $${\alpha }_{0}={\mu }_{B}/(eD)$$, where μB is the Bohr magneton, D is the diffusion coefficient, e is the electron charge and we use theoretical units $$c=\hslash ={k}_{B}=1$$. Usually the orbital effect in superconductors dominates over the paramagnetic one, provided that the second critical field Hc2 is not too high so that $${\mu }_{B}{H}_{c2}\ll {T}_{c}$$. In this case the Maki parameter is small $${\alpha }_{0}\ll 1$$. Exceptions are the high-field superconductors were the Zeeman shift can become relatively large at fields not exceeding Hc216,18,19,20,21,22. The paramagnetic effect can be significantly enhanced due to the geometrical confinement in thin superconducting films12,14,23,24. Alternatively, the spin splitting in superconductors can be induced by the exchange interaction of conduction electrons with localized magnetic moments, e.g. aligned magnetic impurities25 or in superconductor/metallic ferromagnet hybrid structures26,27,28. Recently, the systems consisting of superconducting films grown on the surfaces of ferromagnetic insulators like EuS13,29,30,31,32 and GdN33 have been fabricated. The exchange field heff in the superconducting film is induced due to the scattering of conductivity electrons from the ferromagnetic insulator interface34. Such systems are currently studied quite actively as the possible platforms for the advanced radiation sensing technology5,35 and quantum computing with Majorana states36.

The most well known paramagnetic effects in spin-singlet superconductors are the first-order transition into the normal state18,19 and the second-order transition into the inhomogeneous superconducting state induced by the spin-splitting field heff. The inhomogeneous state (FFLO) suggested by Fulde, Ferrell25 and Larkin, Ovchinnikov37 is realized in the narrow window of parameters and suppressed by impurities38 which hinders its experimental realizations39. However the first-order transition into the normal state driven by the Zeeman splitting has been detected in thin aluminum films23. In this paper we focus on the more robust nonequilibrium phenomena which generically appear in the presence of any spin-splitting field in the spin-singlet superconductor5. In particular, we consider the film of type-II superconductor which can host Abrikosov vortices. The example of such setup is shown schematically in Fig. 1. It consists of the thin superconducting film deposited on the magnetic insulator which creates spin splitting of the conduction electron subbands in the superconductor due to the effective exchange interaction heff. In addition there is a magnetic field B directed perpendicular to the film plane to create vortices. The total spin splitting field is given by the superposition $${\boldsymbol{h}}={\mu }_{B}{\boldsymbol{B}}+{{\boldsymbol{h}}}_{eff}$$, so the single-particle Hamiltonian becomes $$H={(i\nabla +e{\boldsymbol{A}})}^{2}/(2m)+\hat{{\boldsymbol{\sigma }}}{\boldsymbol{h}}$$, where A is the vector potential and $$\hat{{\boldsymbol{\sigma }}}$$ is the vector of spin Pauli matrices.

Superconductors with the total spin splitting field $${\boldsymbol{h}}={{\boldsymbol{h}}}_{eff}+{\mu }_{B}{\boldsymbol{B}}$$ coming both due to Zeeman shift and internal exchange are characterized by the renormalized Maki parameter $$\alpha ={\alpha }_{0}h/({\mu }_{B}{H}_{c2})$$. It can become large $$\alpha \sim 1$$ if the total spin splitting is close to the paramagnetic depairing threshold $$h\sim {T}_{c}$$. Such strong spin splitting has been recently obtained in superconductor/ferromagnetic insulator proximity structures used for the generation of the long-range spin accumulation in the non-local spin valve geometries5,13,31,32,33. Due to the large exchange field this regime can be achieved even if the Zeeman effect is small, that is when $${\mu }_{B}B\ll {T}_{c}$$.

Although we focus on the superconductor/ferromagnetic insulator bilayer system, the regime when $$\alpha \sim 1$$ is also possible in high-field bulk superconductors where the spin splitting comes solely from the Zeeman effect16,20. Similar behavior can be observed in magnetic superconductors40, such as borocarbides41 where weak ferromagnetic ordering is possible42 so that vortex cores can host localized paramagnetic moments43 and weak pinning facilitates flux-flow regime. The effective spin-splitting can be obtained due to the inverse proximity effect in superconductor/metallic ferromagnet hybrids26,27,28. For instance, recently the flux-flow regime has been realized44 in Nb/PdNi/Nb trilayers which makes it potentially possible to study spin accumulation induced by vortex motion in such systems.

Below we demonstrate that α becomes the only relevant parameter which determines the amplitude of the pure spin current generated by the vortex motion. The latter can be characterized by the spin Hall angle $${\theta }_{sH}=e{j}_{s}/j$$, where js is the induced spin current and and $${\boldsymbol{j}}={{\boldsymbol{j}}}_{t}$$ is the charge current equal to the transport current generated by the external source. The spin Hall angle can be estimated as $${\theta }_{sH}\sim \alpha$$. At the paramagnetic threshold $$h\sim {T}_{c}$$ it can reach $${\theta }_{sH}\sim 1$$ which is much larger than the record values $${\theta }_{sH} < 0.1$$ obtained in the heavy metal spin current generators1.

The above result is rather surprising because the maximal spin splitting $$h\sim {T}_{c}$$ is very small as compared to the Fermi energy $${\varepsilon }_{F}$$, since in usual superconductors $${\varepsilon }_{F}/{T}_{c}\sim {10}^{2}-{10}^{3}$$. In this case the polarization, which is the relative difference between spin-up/down conductivities is rather small ~$$h/{\varepsilon }_{F}\ll 1$$. This limit yields vanishing spin-polarized component of the resistive current. However, it is the vortex motion which generates much larger spin current in the transverse direction $${{\boldsymbol{j}}}_{s}\perp {\boldsymbol{j}}$$. Large values of spin Hall angle are in the sharp contrast with extremely small ordinary (charge) Hall angle θH. As shown by Kopnin45 the Hall angle is determined by the parameter $${\theta }_{H}\sim \tau {T}_{c}^{2}/{\varepsilon }_{F}$$, where $$\tau$$ is the electronic mean free time due to the scattering by impurities. The first factor $$\tau {T}_{c}\ll 1$$ in the dirty regime while the second factor $${T}_{c}/{\varepsilon }_{F}\ll 1$$ in all superconductors. So, $${\theta }_{H}\ll 1$$ is determined by the product of two very small parameters, unlike the spin Hall angle which does not contain any small parameters. Thus, the charge and spin transport channels are separated in this system and the spin current generated by the vortex motion is to the very high accuracy the pure one with only vanishingly small admixture of the transverse charge current. Simultaneously, for $${\theta }_{H}\ll 1$$ vortices move almost perpendicularly to the transport current $${{\boldsymbol{v}}}_{L}\perp {{\boldsymbol{j}}}_{t}$$ according to the textbook picture of the flux-flow effect which we use in the discussion below.

The scheme of the flux-flow direct spin Hall effect is shown in Fig. 1a. Here, we assume that the superconductor with spin-splitting field and vortices is subject to the transport charge current j generated by the external source. This transport current induces the Lorenz force acting on the vortex lines in the direction perpendicular to current $${{\boldsymbol{F}}}_{L}\propto {\boldsymbol{j}}\times {\boldsymbol{B}}$$. Provided that the Lorenz force overcomes the pinning barrier, vortices start to move in the transverse direction with the velocity $${{\boldsymbol{v}}}_{L}\perp {\boldsymbol{j}}$$. Taking into account the spin polarization Sloc which exists inside each vortex core due to paramagnetic response, this motion generates the transverse pure spin current $${{\boldsymbol{j}}}_{s}\approx {n}_{v}{{\boldsymbol{v}}}_{L}{S}_{loc}$$, where $${n}_{v}=B/{\varphi }_{0}$$ is the vortex density, $${\varphi }_{0}$$ is flux quantum.

Vortex cores in diffusive superconductors can be though of as the normal metal tubes, of the diameter determined by the coherence length $$\xi$$. In the presence of spin splitting field, the vortex cores contain localized spin $${S}_{loc}\sim {\chi }_{n}h{\xi }^{2}$$ per unit vortex length, where $${\chi }_{n}={N}_{0}$$ is the normal metal paramagnetic susceptibility and N0 is the Fermi-level density of states. To estimate js we substitute the flux-flow vortex velocity $${v}_{L}=-\,E/B$$ and get $${\theta }_{sH}\sim h/(eD{H}_{c2})\sim \alpha$$, so that α appears to be the only small parameter limiting the spin current generation. The physical reason for large θsH lies in the fast motion of vortices which can be compared e.g. with the Drude-model electron drift velocity $$\bar{v}={\sigma }_{n}E/(ne)$$, where the conductivity is $${\sigma }_{n}={e}^{2}{N}_{0}D$$. At $$B\approx {H}_{c2}$$ we have the relation $${v}_{L}\approx \bar{v}{\varepsilon }_{F}/{T}_{c}\gg \bar{v}$$. Therefore spin polarization can be transported much faster by moving vortices than by electrons drifting along the electric field.

Along with the direct SHE we propose also the scheme of the inverse flux-flow SHE shown in Fig. 1b. The mechanism is based on the injection of spin-polarized quasiparticle current into the superconductor by applying the voltage through the spin-filtering ferromagnetic electrodes with polarization P. The resulting spin-dependent voltage Vs generates the spatially-inhomogeneous non-equilibrium spin accumulation which we hereafter denote μz. Its gradient $$\nabla$$μz will be shown to produce the longitudinal force acting on the spin-polarized vortex cores pushing them towards one of the ferromagnetic electrodes. The vortex lattice motion with velocity vL generates electric field in the transverse direction $${\boldsymbol{E}}\parallel {\boldsymbol{B}}\times \nabla {\mu }_{s}$$ thus providing the novel mechanism of inverse SHE.

## Model

To quantify these effects we use the framework of Keldysh-Usadel theory46,47 describing the spin current and spin accumulation induced by the vortex motion in the usual s-wave spin-singlet superconductor in the diffusive regime5. We consider the range of magnetic fields close to Hc2, neglecting screening and using the Abrikosov solution for the moving vortex lattice. We will show that in addition to the large average spin current there is also the oscillating part which can be considered as the high-frequency source of the spin current at the nearly-terahertz range48.

We use the formalism of quasiclassical Green’s functions (GF)46,47 generalized to describe the non-equilibrium spin states in diffusive superconductors5,49, $$\breve{g}=(\begin{array}{cc}{\hat{g}}^{R} & {\hat{g}}^{K}\\ 0 & {\hat{g}}^{A}\end{array})$$, where $${\hat{g}}^{R/A/K}$$ are the retarded/advanced/Keldysh components which are the matrices in spin-Nambu space and depend on two times and a single spatial coordinate variable $$\breve{g}=\breve{g}({t}_{1},{t}_{2},{\boldsymbol{r}})$$. Choosing the z-axis in spin space to be directed along the spin-splitting field h we consider general expressions for the spin density deviation from the normal state one, $$S=S(t,{\boldsymbol{r}})$$, and spin current $${\boldsymbol{j}}={{\boldsymbol{j}}}_{s}(t,{\boldsymbol{r}})$$ determined through the GF at coinciding coordinates $${t}_{1,2}=t$$

$$S=-\,\frac{\pi {\chi }_{n}}{8}{\rm{Tr}}[{\hat{\tau }}_{3}{\hat{\sigma }}_{3}{\hat{g}}^{K}]$$
(1)
$${{\boldsymbol{j}}}_{s}=\frac{\pi {\sigma }_{n}}{8{e}^{2}}{\rm{Tr}}[{\hat{\sigma }}_{3}{(\hat{g}\circ {\hat{\partial }}_{{\boldsymbol{r}}}\hat{g})}^{K}]$$
(2)

Here after $${\hat{\sigma }}_{i}$$, $${\hat{\tau }}_{i}$$ are the Pauli matrices in spin and Nambu spaces, and the symbolic time-convolution operator is given by $$(A\circ B)\,({t}_{1},{t}_{2})=\int \,dtA({t}_{1},t)B(t,{t}_{2})$$, the covariant differential superoperator is defined by $${\hat{\partial }}_{{\boldsymbol{r}}}=\nabla -ie{[{\hat{\tau }}_{3}{\boldsymbol{A}}]}_{t}$$ and the two-time commutator is defined as $${[X,g]}_{t}=X({t}_{1})g({t}_{1},{t}_{2})-g({t}_{1},{t}_{2})X({t}_{2})$$, similarly for anticommutator $${\{,\}}_{t}$$. The Keldysh GF is conveniently described using the parametrization $${\hat{g}}^{K}={\hat{g}}^{R}\circ \hat{f}-\hat{f}\circ {\hat{g}}^{A}$$ which follows from the normalization condition $$\breve{g}\circ \breve{g}=\delta ({t}_{1}-{t}_{2})$$. Here $$\hat{f}=\hat{f}({t}_{1},{t}_{2},{\boldsymbol{r}})$$ is the generalized distribution function. For calculations we use mixed representation $$\breve{g}({t}_{1},{t}_{2})={\int }_{-\infty }^{\infty }\,\breve{g}(\varepsilon ,t){e}^{-i\varepsilon ({t}_{1}-{t}_{2})}d\varepsilon /2\pi$$, where $$t=({t}_{1}+{t}_{2})/2$$ is the ‘center of mass’ time.

In the flux-flow regime we assume that vortices move with the constant velocity vL. In the zero-order approximation the distribution function is equilibrium $$\hat{f}(\varepsilon )={f}_{0}(\varepsilon ){\hat{\tau }}_{0}\equiv \,\tanh \,[\varepsilon /(2T)]{\hat{\tau }}_{0}$$. Similarly, the spectral functions have their equilibrium forms in the frame moving together with vortices $${\hat{g}}^{R/A}({\boldsymbol{r}})\approx {\hat{g}}_{0}^{R/A}({\boldsymbol{r}}-{{\boldsymbol{v}}}_{L}t)$$. This approximation yields zero spin current which is absent in equilibrium spin-singlet superconductors. Thus, we need to consider corrections in the linear-response regime which is realized provided the vortex velocity vL is small enough to neglect Joule heating, pair breaking or vortex-core shrinking effects50,51. For this purpose we take into account first-order terms in the gradient expansion of time convolutions52,53 as well as the non-equilibrium corrections to the spectral functions $${\hat{g}}_{ne}^{R/A}$$ and the distribution function $${\hat{f}}_{ne}=\hat{f}-{f}_{0}{\hat{\tau }}_{0}$$.

The nonequilibrium GF is determined by the Keldysh-Usadel equation46,47 which should be solved together with the self-consistency equations. In general this problem is very complicated and has never been approached even numerically. However, the regime of high magnetic fields $${H}_{c2}-B\ll {H}_{c2}$$ allows for significant simplifications based on the existence of the Abrikosov vortex lattice solution for the superconducting order parameter. In this case it is possible to find analytically nonequilibrium corrections to the spectral functions $${\hat{g}}^{R/A}$$ and the components of distribution function $$\hat{f}$$. First of all, we employ the analytical expression for the order parameter distribution in the moving vortex lattice. Assuming the particular directions of vortex velocity $${{\boldsymbol{v}}}_{L}={v}_{L}{\boldsymbol{y}}$$ and electric field $${\boldsymbol{E}}=E{\boldsymbol{x}}$$ we choose the time-dependent vector potential in the form $${\boldsymbol{A}}=Bx{\boldsymbol{y}}-Et{\boldsymbol{x}}$$. Then the order parameter is given by superposition of the first Landau-level nuclei $${\mathcal L} (x)=\exp (\,-\,{x}^{2}/2{L}_{H}^{2})$$, so that $${\rm{\Delta }}={b}_{0}{e}^{-2ieEtx}\,{\sum }_{n}\,{C}_{n}{e}^{inp(y-{v}_{L}t)} {\mathcal L} (x-n{x}_{0})$$. Here b0 is magnetic field-dependent amplitude derived in the Supplementary Material, $${x}_{0}=p{L}_{H}^{2}$$ determines the distance between neighbour superconducting nuclei and $${L}_{H}=1/\sqrt{2e{H}_{c2}}$$ is the magnetic length. For the triangular lattice $${C}_{n+1}={e}^{i{(-1)}^{n}\pi /4}$$, $$p{L}_{H}=\sqrt{\pi \sqrt{3}}$$ and for the square one $${C}_{n}=1$$, $$p{L}_{H}=\sqrt{2\pi }$$.

Second, we use the known solutions for the equilibrium spectral functions in the vortex lattice near the upper critical field54. Here we take into account the spin-splitting field by shifting the quasiparticle energies according to $${\varepsilon }_{\sigma }=\varepsilon -\sigma h$$, where $$\sigma =\pm$$. Then the spin-up $${g}_{0+}^{R}$$ and spin-down $${\hat{g}}_{0-}^{R}$$ GFs are given by

$${\hat{g}}_{0\sigma }^{R}({\boldsymbol{r}},\varepsilon )=[1+\frac{|{\rm{\Delta }}{|}^{2}}{2{(iq+{\varepsilon }_{\sigma })}^{2}}]\,{\hat{\tau }}_{3}+\frac{i|{\rm{\Delta }}|{\hat{\tau }}_{2}{e}^{-i\phi {\hat{\tau }}_{3}}}{iq+{\varepsilon }_{\sigma }},$$
(3)

and $${\hat{g}}_{0}^{A}=-\,{\hat{\tau }}_{3}{\hat{g}}_{0}^{R\dagger }{\hat{\tau }}_{3}$$ for the advanced GF. Here $$q=e{H}_{c2}D$$ and the order parameter is $${\rm{\Delta }}=|{\rm{\Delta }}|{e}^{i\phi }$$. The total GF is given by $${\hat{g}}_{0}^{R}={\hat{\sigma }}_{0}({\hat{g}}_{0+}^{R}+{\hat{g}}_{0-}^{R})/2+{\hat{\sigma }}_{3}({\hat{g}}_{0+}^{R}-{\hat{g}}_{0-}^{R})/2$$.

These spin-polarized spectral functions provide the description of equilibrium spin density modulation in a superconductor with spin-splitting field in the presence of vortex lattices. The periodic spin density patterns calculated for the typical cases of triangular and square lattices are shown in the Fig. 2. The spin polarization demonstrates enhancement at the vortex cores and suppression between vortices where the order parameter is larger. Thus even in the regime of dense vortex lattices there is an excess spin polarization Sloc localized in the vortex cores. It is natural to expect that the motion of such spin-polarized vortices will produce pure spin currents. Below we demonstrate the presence of these spin currents by an explicit calculation in the flux-flow regime considering the non-equilibrium situation when the vortex lattice moves under the action of the transport current jt. We will calculate the spin current density induced by the vortex motion as well as the non-equilibrium spin accumulation and charge imbalance near the vortex cores.

## Results

### Spin current

The expression for spin current (2) can be decomposed into the parts js1 related to the distortions and time derivatives of spectral functions $${\hat{g}}^{R/A}$$ and js2 which is determined by the corrections to the distribution function $${\hat{f}}_{ne}=\hat{f}-{f}_{0}$$, where f0 is the equilibrium distribution function. The first part of the spin current js1 is determined by the non-equilibrium corrections to the spectral quantities while it contains only the equilibrium distribution function. In the charge sector these corrections yield the Caroli-Maki part of the flux-flow conductivity55. The important difference is that the charge current is determined by the corrections induced by the order parameter distortions in the moving vortex lattice while they do not contribute to the spin current. Since we neglect the spin-flip and spin-orbital scattering processes the total spin density and the spin current given by the Eqs (1) and (2) satisfy the continuity equation $${\partial }_{t}S+\nabla \cdot {{\boldsymbol{j}}}_{s}=0$$. When implementing the linear response calculations we collect all contributions to the spin current to satisfy the continuity equation to the first order in vortex velocity. The derivation of this conservation law is discussed in the Supplementary Material.

The distortions of spectral GF generated by the vortex motion can be found using (3) and the normalization condition expanded in the mixed representation up to the first-order time derivatives. After performing several steps of analytical calculations described in Supplementary Material we obtain the spectral-related part of the spin current

$${{\boldsymbol{j}}}_{s1}=-\,\frac{{\sigma }_{n}}{16{e}^{2}}\frac{{\mathrm{Im}{\rm{\Psi }}}^{(2)}}{{(\pi T)}^{2}}{\rm{Re}}[{\rm{\Delta }}{(\hat{{\rm{\Pi }}}{\partial }_{t}{\rm{\Delta }})}^{\ast }-{\partial }_{t}{\rm{\Delta }}{(\hat{{\rm{\Pi }}}{\rm{\Delta }})}^{\ast }],$$
(4)

where $${\rm{\Psi }}={\rm{\Psi }}[1/2+(q+ih)/(2\pi T)]$$ is digamma function, $${{\rm{\Psi }}}^{(n)}(z)={d}^{n}{\rm{\Psi }}(z)/d{z}^{n}$$ and $$\hat{{\rm{\Pi }}}=\nabla -2ie{\boldsymbol{A}}$$. This part of the spin current has the non-zero space- and time-average $$\langle {{\boldsymbol{j}}}_{s}\rangle =\langle {{\boldsymbol{j}}}_{s1}\rangle$$. Indeed, as we show below the second part of the spin current related to the non-equilibrium distribution function does not contribute to the average $$\langle {{\boldsymbol{j}}}_{s2}\rangle =0$$.

At low temperatures, the space- average spin current can be written in the intuitively transparent expression

$$\langle {{\boldsymbol{j}}}_{s}\rangle =-\,{{\boldsymbol{v}}}_{L}\langle S\rangle (\frac{2}{1+{\alpha }^{2}})$$
(5)

where $$\langle S\rangle$$ is the average spin density deviation from the normal state. The absolute magnitude of $$\langle S\rangle$$ is determined by the average order parameter amplitude $$\langle {{\rm{\Delta }}}^{2}\rangle$$ which can be found analytically using the expression for the order parameter amplitude near the upper critical field56,57. Then, substituting $$\langle S\rangle$$ derived in the Supplementary Material and taking into account that the vortex velocity is $${v}_{L}=-\,j/({\sigma }_{n}{H}_{c2})$$ we obtain the following analytical expression for the spin Hall angle $${\theta }_{sH}=e\langle {{\boldsymbol{j}}}_{s}\rangle /{{\boldsymbol{j}}}_{t}$$, as a function of the average magnetic induction at low temperatures,

$${\theta }_{sH}(B)=-\,\frac{4\alpha }{{\beta }_{L}(1-{\alpha }^{4})}(1-\frac{B}{{H}_{c2}}),$$
(6)

where the Abrikosov parameter equals $${\beta }_{L}=1.16$$ for the triangular and $${\beta }_{L}=1.18$$ for the square lattice58. The growth of θsH(B) with decreasing B given by Eq. (6) close to Hc2 should continue at lower fields until the order parameter between vortices becomes fully developed at $$B\approx 0.3{H}_{c2}$$. In this regime $${\theta }_{sH}\propto \alpha$$ without any small parameters so that $${\theta }_{sH}\sim 1$$ for large exchange splitting $$h\sim {T}_{c}$$. Besides that, according to Eq. (6) large spin Hall angle can be obtained already in the regime $$\mathrm{(1}-B/{H}_{c2})\ll 1$$ provided that $$1-\alpha \ll 1-B/{H}_{c2}$$. Note that we restrict our consideration to $$\alpha < 1$$ when the superconducting transition at $$B={H}_{c2}$$ is of the second order20,59.

Now let us consider the second part of the spin current determined by the correction to the distribution function. Due to the smallness of the order parameter near Hc2 it can be written as $${{\boldsymbol{j}}}_{s2}=({\sigma }_{n}/{e}^{2})\nabla {\mu }_{s}$$, where $${\mu }_{s}={\int }_{-\infty }^{\infty }\,{f}_{T3}(\varepsilon )d\varepsilon /2$$ is the quantity which can be considered as the spin-dependent shift of the chemical potential and $${f}_{T3}={\rm{Tr}}[{\hat{\sigma }}_{3}\,\hat{f}]/4$$ is the spin-dependent component of the distribution function5,11. Besides fT3 the vortex motion excites the electron-hole imbalance described by the component of distribution function $${f}_{T}={\rm{Tr}}[{\hat{\tau }}_{3}\,\hat{f}]/4$$. Both these components are determined by the following kinetic equations (see Supplementary Material for derivation)

$$D{\nabla }^{2}{f}_{T3}=-\,{\partial }_{\varepsilon }\,{f}_{0}\{e{{\boldsymbol{J}}}_{se}{\boldsymbol{E}}+\frac{{\rm{Tr}}[{\hat{\sigma }}_{3}{\partial }_{t}\hat{{\rm{\Delta }}}({\hat{g}}_{0}^{R}-{\hat{g}}_{0}^{A})]}{8}\}$$
(7)
$$D{\nabla }^{2}{f}_{T}=-\,{\partial }_{\varepsilon }\,{f}_{0}\{e\nabla ({{\mathscr{D}}}_{T}{\boldsymbol{E}})-\frac{{\rm{Tr}}[{\hat{\tau }}_{3}{\partial }_{t}\hat{{\rm{\Delta }}}({\hat{g}}_{0}^{R}+{\hat{g}}_{0}^{A})]}{8}\},$$
(8)

where $${{\mathscr{D}}}_{T}=D{\rm{Tr}}({\hat{\tau }}_{0}{\hat{\sigma }}_{0}-{\hat{\tau }}_{3}\,{\hat{g}}_{0}^{R}{\hat{\tau }}_{3}\,{\hat{g}}_{0}^{A})/8$$ is the diffusion coefficient for the charge imbalance modified by superconducting correlations, $${{\boldsymbol{J}}}_{se}=D{\rm{Tr}}[{\hat{\tau }}_{3}{\hat{\sigma }}_{3}({\hat{g}}_{0}^{R}\hat{\nabla }{\hat{g}}_{0}^{R}-{\hat{g}}_{0}^{A}\hat{\nabla }{\hat{g}}_{0}^{A})]/8$$ is spectral spin-energy current density11 and the gap operator in the second term of the r.h.s. is $$\hat{{\rm{\Delta }}}=i|{\rm{\Delta }}|{\hat{\tau }}_{2}{e}^{-i\phi {\hat{\tau }}_{3}}$$. For the general-form Abrikosov vortex lattice, Eqs (7) and (8) can be solved analytically yielding the expression which can be found in the Supplementary Material. Since fT3 and hence μs have to be periodic functions, the contribution to spin current js2 has zero space and time-averages but contributes to the AC component of js.

The overall distributions spin currents are shown in Fig. 3 produced using the Matplotlib package60 for two different vortex lattice geometries. Here one can see that the spin current mostly flows along the vortex chains with maximal current concentrated in the vortex cores. This result confirms our initial qualitative picture shown in Fig. 1 that the spin is transported by the moving spin-polarized vortex cores. In addition, in Fig. 3C,D one can see a non-trivial distribution of the spatially-periodic part of the current $${\tilde{j}}_{s}={{\boldsymbol{j}}}_{s}-\langle {{\boldsymbol{j}}}_{s}\rangle$$, which is important for the AC spin current generation discussed below. The periodic part $${\tilde{{\boldsymbol{j}}}}_{s}$$ forms two standing eddies localized close to the vortex core similar to that which are formed by the low-Reynolds viscous flow past a cylinder.

### Spin accumulation and charge imbalance

Besides generating the spin current, moving vortices produce other types of non-equilibrium states in the superconductor, such as the charge imbalance and the non-equilibrium spin accumulation which we denote as $$\tilde{\mu }$$ and $${\tilde{\mu }}_{s}$$, respectively. These quantities have been widely used as the experimentally observable characteristics of the non-equilibrium superconducting states both in spin -degenerate61,62,63,64,65,66,67,68 and spin-split systems12,13,14,15,24,31,69. General expressions for charge imbalance and spin accumulation in terms of the quasiclassical GF read as

$$\tilde{\mu }=-\,\frac{\pi }{8}{\rm{Tr}}{\hat{g}}_{ne}^{K}(t,t,{\boldsymbol{r}})$$
(9)
$${\tilde{\mu }}_{s}=-\,\frac{\pi }{8}{\rm{Tr}}[{\hat{\tau }}_{3}{\hat{\sigma }}_{h}\,{\hat{g}}_{ne}^{K}]\,(t,t,{\boldsymbol{r}})$$
(10)

where $${\hat{g}}_{ne}^{K}$$ is non-equilibrium part of Keldysh GF.

In contrast to the stationary cases considered before5,49 in the time-dependent system these imbalances are determined not only by the the nonequlibrium distribution function components $$\mu =\int \,{f}_{T}d\varepsilon /2$$ and μs discussed above. Besides that they contain additional terms determined by the dynamics of the order parameter

$${\tilde{\mu }}_{s}=-\,{\mu }_{s}+{v}_{L}{\partial }_{y}|{\rm{\Delta }}{|}^{2}\frac{{\mathrm{Im}{\rm{\Psi }}}^{(2)}}{{(4\pi T)}^{2}}$$
(11)
$$\tilde{\mu }=-\,\mu -{v}_{L}{\partial }_{x}|{\rm{\Delta }}{|}^{2}\frac{{\mathrm{Re}{\rm{\Psi }}}^{(2)}}{2{(4\pi T)}^{2}},$$
(12)

see the Supplementary Material for details of the derivation. Distributions of $$\tilde{\mu }$$ and $${\tilde{\mu }}_{s}$$ generated by the moving triangular and square vortex lattices are shown in Fig. 4. The patterns of charge imbalance agree with the qualitative picture suggested by Bardeen and Stephen70 where the vortex motion is accompanied by the generation of dipolar-like electric field near the vortex core, corresponding to the electric dipole directed perpendicular to the vortex velocity vL. On the contrary, the “spin dipoles” corresponding to the patterns of $${\tilde{\mu }}_{s}$$ are directed along vL. Note also, that spin accumulation is proportional to the generalized Maki parameter, while $$\tilde{\mu }$$ remains finite when $$\alpha \to 0$$. These quantities $$\tilde{\mu }$$ and $${\tilde{\mu }}_{s}$$ can be directly measured with the help of ferromagnetic detector electrodes5,12,13,14,15,24,31,62,65,67,68,69 thus providing experimental tool to detect the motion of individual vortices.

### Flux-flow inverse spin Hall effect (ISHE)

We suggest the new mechanism of the flux-flow ISHE which is based on the previously unknown effect of longitudinal vortex motion driven by the spin current or spin accumulation injected into the superconductor from the attached ferromagnetic electrodes with polarization P. We denote Vs the corresponding spin-dependent external bias. For simplicity we assume that the polarization is aligned with the spin-splitting filed in the superconductor $${\boldsymbol{P}}\parallel {\boldsymbol{h}}$$. To calculate the force acting on vortex from the injected spin current we consider the regime of temperatures close to the critical one Tc. In this case we can neglect the superconducting corrections to the density of states. This assumption simplifies expression for spin-dependent part of the distribution function which can be taken in the form corresponding to the normal metal $${f}_{T3}({\boldsymbol{r}})=-\,{\mu }_{s}({\boldsymbol{r}}){\partial }_{\varepsilon }\,{f}_{0}$$ where $${\mu }_{s}={\mu }_{s}({\boldsymbol{r}})$$ is the spatially-inhomogeneous spin accumulation generated by the external bias Vs. Besides that here we consider the regime of small fields $$B\ll {H}_{c2}$$ when vortices can be considered as individual objects. The force acting on the single vortex from non-equilibrium spin-polarized environment Fd can be calculated using the known general expression52,53. Near the critical temperature when $$|{\rm{\Delta }}|\ll T$$ we obtain the simple analytical result $${{\boldsymbol{F}}}_{d}\approx \nabla {\mu }_{s}{S}_{loc}|{\rm{\Delta }}{|}^{2}/{T}_{c}^{2}$$, where Sloc is the total spin localized in the vortex core. This driving force, balanced by the friction $${{\boldsymbol{F}}}_{v}=-\,\rho {{\boldsymbol{v}}}_{L}$$, where $$\rho$$ is the vortex viscosity coefficient, yields the flux-flow velocity $${{\boldsymbol{v}}}_{L}\parallel \nabla {\mu }_{z}$$. Its absolute value can be found using the known analytical expression for viscosity coefficient $$\rho ={\varphi }_{0}{\sigma }_{n}\beta {H}_{c2}$$. The temperature dependence close to Tc is determined by the coefficient $$\beta ={\beta }_{0}/\sqrt{1-T/{T}_{c}}$$, where β0 is some numerical value71,72,73. Taking into account that the concentration of vortices is determined by the average magnetic induction B and using the usual expression for the sample-average electric field $${\boldsymbol{E}}=-\,{{\boldsymbol{v}}}_{L}\times {\boldsymbol{B}}$$ we obtain the relation

$$\frac{eE}{\nabla {\mu }_{s}}\approx \frac{h}{q}\frac{|{\rm{\Delta }}{|}^{2}}{{T}_{c}^{2}}\frac{B}{{H}_{c2}}\sqrt{1-\frac{T}{{T}_{c}}}.$$
(13)

The obtained result (13) yields the linear response relation for the inverse spin Hall effect because the electric field E and the corresponding electric current are generated in response to the applied spin-dependent voltage Vs. The overall temperature dependence of the generated electric field $$E\propto {(1-T/{T}_{c})}^{3/2}$$ is determined by the order parameter amplitude $${{\rm{\Delta }}}^{2}\propto (1-T/{T}_{c})$$ and the additional factor which comes from the divergence of vortex viscosity coefficient close to the critical temperature72 $$\rho \propto 1/\sqrt{1-T/{T}_{c}}$$.

## Discussion and Conclusions

We have found the spin current generation by moving vortices which penetrate the whole volume of the type-II superconductor. Thus the obtained spin current in contrast to the previously known schemes based on the injection mechanisms exists everywhere in the sample volume and is prone to the spin relaxation mechanisms such as the spin-flip scattering. The predicted spin current generation can be tested in the open circuit geometries when the vortices annihilate at the insulating boundary. In this case the net spin current at the boundary, $$y=0$$, should vanish $${j}_{sy}(y=0)=0$$ generating the surface spin accumulation $${V}_{s}={\mu }_{s}/e$$ which can be measured by the ferromagnetic detector electrodes12,13,14,15.

The second possible experimental test is based on the direct measurement of the spin current injected through superconductor interfaces into the inverse spin Hall detector74,75. This approach allows for the measurement of both the DC and the high-frequency AC spin current signals. The latter is generated due to the periodic structure of moving vortex lattice. The distribution of the space-periodic spin current component is shown in Fig. 3C,D. The amplitude of AC component flowing through the superconductor interface, $${\langle {\tilde{j}}_{sy}\rangle }_{x}$$, is determined by the variations of the current average along the boundary, $${\langle {j}_{sy}\rangle }_{x}$$, with respect to the constant background current $$\langle {j}_{s}\rangle$$. At low temperatures, the relative magnitude is given by $${\langle {\tilde{j}}_{sy}\rangle }_{x}/\langle {j}_{s}\rangle =(1-{\langle {{\rm{\Delta }}}^{2}\rangle }_{x}/\langle {{\rm{\Delta }}}^{2}\rangle )\,(1+{\alpha }^{2})/2$$. According to the recent measurements in Pb, the frequency of vortex entry into the superconducting sample can reach dozens of gigahertz48 and the THz range in layered high-temperature superconductors76. In the suggested setup this is the frequency of the AC spin current generated by the vortex motion. The high-frequency spin current generation can be useful in antiferromagnetic spintronics characterized by the terahertz-range dynamics of the magnetic system77.

The charge imbalance and spin accumulation have been accessed experimentally using non-local conductance measurements12,13,14,15,24,31,62,65,67,68,69, when the non-equilibrium states were created by the current in the injector circuit. The non-local electric signal has been measured between the normal detector electrodes, either ferromagnetic or non-ferromagnetic attached to the different points of superconducting sample. Here we show that in the flux-flow regime the non-equilibrium states with non-zero charge imbalance $$\tilde{\mu }$$ and spin accumulation $${\tilde{\mu }}_{s}$$ appear in the absence of quasiparticle injector current, but rather just due to the vortex motion. The quantities $$\tilde{\mu }$$ and $${\tilde{\mu }}_{s}$$ can be measured using the same electrical detection circuits as in the non-local conductance measurement setups. For example, the tunneling current at the non-ferromagnetic normal detector electrode is proportional to $$\tilde{\mu }$$. In case of the ferromagnetic electrode there is a contribution to the detector current5 proportional to $${\tilde{\mu }}_{s}$$. In the flux-flow regime each vortex carries the distributions of $$\tilde{\mu }$$ and $${\tilde{\mu }}_{s}$$ localized in the vortex core. Thus, moving vortices passing close to the detector electrode are expected to generate pulses of the tunneling current or voltage, depending on the detection scheme. This provides a tool capable for detecting the motion of individual vortices. In contrast to the magnetometer techniques it does not have the frequency limitations48 and therefore can directly resolve the ultrafast vortex motion with the frequencies up to the dozens of gigahertz.

To conclude, we have demonstrated fundamental mechanisms of direct and inverse spin Hall effects due to the flux-flow of Abrikosov vortices in type-II superconductors with spin-splitting field. The spin splitting can be generated by the adjacent ferromagnetic insulator as shown in Fig. 1, by the Zeeman effect in the magnetic field applied in the plane of thin superconducting film or due to the inverse proximity effect in superconductor/ferromagnet hybrid structures26,27,28. The pure spin current carried by the fast vortices moving in the transverse direction is characterized by the large spin Hall angle which in general does not contain any small parameters. Besides that there is also an AC component which appears due to the periodic structure of the vortex lattice. The AC spin current has the same order of magnitude as the average one. This effect can be used for the generation of spin signals in wide frequency domain up to the range of therahertz. We pointed out the longitudinal driving force exerted on vortex by the injected spin current. The vortex motion generated by this force leads to the inverse spin Hall effect. This mechanism can be applied for flux-flow based detection of pure spin currents.

## Data Availability

No datasets were generated or analysed during the current study.

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## Acknowledgements

This work was supported by the Academy of Finland. It is our pleasure to acknowledge discussions with Jan Aarts, Tero T. Heikkilä and Alexander Mel’nikov.

## Author information

Authors

### Contributions

Both authors, A.V. and M.S. contributed equally to calculations and writing of the manuscript.

### Corresponding author

Correspondence to Mikhail Silaev.

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Vargunin, A., Silaev, M. Flux flow spin Hall effect in type-II superconductors with spin-splitting field. Sci Rep 9, 5914 (2019). https://doi.org/10.1038/s41598-019-42034-y

• Accepted:

• Published:

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text:
# ArcShoka

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$u\!+\!\mathrm i v= \mathrm{ArcShoka}(x\!+\!\mathrm i y)$

ArcShoka function is inverse function of Shoka function and Abel function for the Keller function;

$\displaystyle \mathrm{ArcShoka}(z)= z + \ln\!\left( \frac{\mathrm e^z-1}{\mathrm e-1} \right)$

Warning! Yet, this article is very draft

## Various inverse functions

Various ways of inversion of the Shoka function lead to the same behavior along the real axis, in the same way as the Shoka function and the Shoko function have identical behavior in vicinity of the real axis; but but different directions of the cutlines.

For efficient comparison of various superfunctions $F$ of the transfer functions $T$ of realistic physical systems, the extension to the complex plane seems to be important tool. This allows to specify the unique solution of the Transfer equation

$T(F(z))=F(z\!+\!1)$

without to refer to the specific way of evaluation of superfunction (although it seems, in many cases of the laser science, the physically-meaningful superfunction can be constructed with the regular iteration). For this reason, the special article about function ArcShoka is loaded. One of goals of this article is comparison of various representations for the ArcShoka function and postulating it as the definition.

The ArcShoka shown in the complex map at right seems to be the most convenient for comparison with the Abel function of similar transfer functions for simple models of the optical media.

## Behavior along the real axis

For real values of the argument, there is no need to make difference between Shoka function and Shoko function, and no need to make any difference between their inverse functions (providing that the cutline do not cross the real axis.) The explicit plot at the picture could be obtained also by the reflection of the plot for the Shoka function from the angle bisector of the First quadrant (id est, line $y=x$). However, for the testing and debugging of the algorithms for the construction of superfunctions and the Abel function, it is important to have the detailed descriptions for the case, when these functions can be expressed through the elementary functions.

## ArcShoka as Abel function

The ArcShoka function is Abel function for the transfer function $T=\mathrm{Keller}$; the Keller function can be defined with

$\mathrm{Keller}(z)=\ln\!\Big(1+\mathrm e(\mathrm e^z\!−\!1)\Big)$

at least in vicinity of the real axis. Then , the $G=\mathrm{ArcShoka}$ satisfies the Abel equation

$G(T(z))=G(z)+1$

format: plain


text:
# Time and Work-Basic test 2

1)

If  1 man or 2 women or 3 boys can finish a work in 88 days, then how many days will 1 man, 1 woman and 1 boy together take to finish the same work?

[Bank Clerks 2009]

2)

A alone can complete a work in 6 days & B alone can complete the same work in 12 days. In how many days can A & B together complete the same work?

[Bank Clerks 2009]

3)

A and B can do a piece of work in 72 days . B and C can do it in 120 days. A and C can do it in 90 days. In what time can A alone do it ?

[SSC 2011]

4)

A takes twice as much time as B & C takes thrice as much time as B to finish a project. Working together, they can finish the project in 12 days. Find the number of days needed for A to do the project alone.

[SSC 2011]

5)

6 boys can complete a piece of work in 16 hrs. In how many hours will 8 boys complete the same work ?

[Bank Clerks 2011]


text:
# Are dimensionality reduction techniques useful in deep learning

I have been working on machine learning and noticed that most of the time, dimensionality reduction techniques like PCA and t-SNE are used in machine learning, but I rarely noticed anyone doing it for deep learning projects. Is there a specific reason for not using dimensionality reduction techniques in deep learning?

It highly depends on your task, your data and your network. Basically, PCA is a linear transformation of the current features. Suppose your data are images or a kind of data that locality is important. If you use PCA you are throwing away those locality information. Consequently, it is clear that people usually do not use them in convolutional networks. For sequential tasks, again it highly depends on your agent whether is online or not. If it is online, you don't have the entire signal from the beginning. Even if you have that for offline tasks, by doing such diminishing transformations you are again throwing away sequential information, I have to say I've not seen the use of them. I guess their main use is in tasks where your problem can be solved using simple MLPs which you don't keep sequential or local information. In those tasks due to the fact that you can employ PCA which leads to the reduction of highly correlated features, the number of parameters of your training model can be reduced significantly.


text:
MathSciNet bibliographic data MR425956 55D15 (58A10) Bousfield, A. K.; Gugenheim, V. K. A. M. On ${\rm PL}$${\rm PL}$ de Rham theory and rational homotopy type. Mem. Amer. Math. Soc. 8 (1976), no. 179, ix+94 pp. Article

For users without a MathSciNet license , Relay Station allows linking from MR numbers in online mathematical literature directly to electronic journals and original articles. Subscribers receive the added value of full MathSciNet reviews.


text:
# Ultra-precise holographic beam shaping for microscopic quantum control

High-resolution addressing of individual ultracold atoms, trapped ions or solid state emitters allows for exquisite control in quantum optics experiments. This becomes possible through large aperture magnifying optics that project microscopic light patterns with diffraction limited performance. We use programmable amplitude holograms generated on a digital micromirror device to create arbitrary microscopic beam shapes with full phase and amplitude control. The system self-corrects for aberrations of up to several $\lambda$ and reduces them to $\lambda/50$, leading to light patterns with a precision on the $10^{−4}$ level. We demonstrate aberration-compensated beam shaping in an optical lattice experiment and perform single-site addressing in a quantum gas microscope for $^{87}\mathrm{Rb}$.


text:
## Basic College Mathematics (10th Edition)

$36\frac{3}{4}$
$42.\frac{7}{8}$ = $42 \times\frac{7}{8}$ = $\frac{42\times7}{8}$ = $\frac{21\times7}{4}$ (divide by common factor 2) = $\frac{147}{4}$ = $36\frac{3}{4}$


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Posted by3 years ago
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## Symbolab a website like Mathway and Wolfram alpha that gives you the steps of the solution for free.

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level 1
Comment deleted3 years ago
level 2
Physics3 points · 3 years ago

Some of these don't work.

level 2
1 point · 3 years ago

Indeed, it gave a wrong answer to a rational function as well -- it suggested crossing a vertical asymptote.

level 1
5 points · 3 years ago

I wonder how many kids will use something like this to cheat on their homework problems...

level 2
31 points · 3 years ago

Honestly, as a student I LOVE sites like these. Not to cheat, but just for the efficiency of having the answer right there.

When I do my homework, I want to know a) if it's right, and b) if not, what errors I am making. A site like this allows me to get those answers AS I do my homework, which I find critical in my ability to process and absorb the material. Having to wait 36 hours or more just to find out if I understood that lesson is unsatisfactory and I feel as it if harms my ability to ingrain the correct information without immediate feedback on my work.

That being said, in nearly all of my classes homework is completion based and a very small percent of my grade. If it were based on accuracy I obviously wouldn't use any outside help.

level 3
[deleted]
2 points · 3 years ago

100% agree, I HATE when there's no quick way to check an answer. The fastest I learn anything is with a solutions manual, and this can act like one for so many things.

level 3
2 points · 3 years ago

Yeah, you're using it about right.

What should be avoided is something like this:

Professor: "Alright students, do problems 1-30 on page 53."

Student: "Okay, so I just need to plug in each of the problems into this website and copy the output and call it done."

In that case, the student is just acting like a glorified copy machine and very little learning is taking place.

level 2
39 points · 3 years ago

It this really any different than going to the library and checking out calculus books that have huge tables of integrals and their solutions? If they're not actually learning from the solutions it's going to bite them in the ass during the exam anyway, but if they are then it's a valuable teaching tool, especially since students can change the integral slightly and see how it affects the solution relatively quickly. The most common questions I get after I do an example in the Calculus class I'm TAing are things like "what would happen if the plus was a minus", "what if that was sine instead of cosine", or "what if the function wasn't multiplied by x."

level 3
6 points · 3 years ago

I first discovered Wolfram Alpha by my stats professor in college telling the class about it. Apparently the department had gotten together to discuss it, and they decided that students would find out anyway and so it was better to point students in the right direction and use it for help than to ignore it and hope students don't cheat.

level 3
1 point · 3 years ago

No, it's not really different. Copying from a book or from a website both have the same danger: Mindlessly copying.

The right way to use an answer section is something like this:

1. Try to solve the problem. Does it match the answer? If yes, carry on.

2. If it doesn't match the answer, go back and try it again. Look for your mistake. Did you find one? What lead you to this mistake? How do you avoid it in the future?

3. If after 5-10 minutes of trying without any success, and you've reviewed the process covered by the instructions and you still don't get it, then you look at the step-by-step solution. Where did you go wrong? Try to take it one step at a time without looking ahead, and get to that point and try to solve the problem from there. Can you do it?

The goal is to get to the point where you can do all of the problems correctly and know that they're correct without checking an answer sheet. You don't have an answer sheet on the exam, and you certainly don't have an answer sheet in real life.

An application like this would be valuable as a teaching tool to show how changing a few parameters changes the final output, in order to build up the mathematical intuition.

level 4
1 point · 3 years ago

I always have wolfram alpha in real life. And I use it pretty frequently, because there are plenty of integrals which it's just not worth my time to solve by hand.

With that said, I do know how to do it by hand, which is important because it gives me intuition about what the answer ought to resemble, and let's me catch any typos I may have made.

level 2
7 points · 3 years ago

My students have been using it for a while. The ones who use it as a guide to solve difficult problems do well. The ones who simply copy it either get expelled or they fail the tests.

level 3
1 point · 3 years ago

They get expelled for using wolfram alpha? What?

Are you allowing the use of such software in the tests or are you telling me they can't use it when studying by themselves?

level 4
3 points · 3 years ago

Earlier in my degree we used to be given take home problems that were a bit harder than the lecturer would expect you to answer in an exam situation. Popping them into wolfram alpha and handing back that answer is just as much plagiarism as copying your report from Wikipedia, suspect that's what OP meant.

level 5
3 points · 3 years ago · edited 3 years ago

Ah, perhaps.

I never know what kind of "help" I can get for that kind of exercises without it being cheating..

Can I use sites like stackexchange? Can I search on the internet for similar examples?

level 6
4 points · 3 years ago · edited 3 years ago

I feel that both of those options are acceptable, but you still need to provide a solution in your own words. In the end, education is about learning the material. It's foolish to while away for hours on a problem only to find that you were just missing a simple step. Other times, the solution may be so difficult that you would have never figured it out. In that case, just explain in greater detail why it works. That indicates to me that you learned the material and invested the time in to understanding it thoroughly.

EDIT: For example, I remember being in analysis and having to find the limit of this complicated function of two variables as (x,y) approached (0,0). I spent over an hour trying all sorts of manipulations before I finally looked it up online. Without reading the full solution, I saw someone mention polar coordinates, and I stopped, tried them, and the solution was simple after that. All it took was someone whispering "polar coordinates" in my ear for me to complete the problem. All of the work was mine, but I needed that simple suggestion to get the answer. Polar coordinates weren't even on my radar because it had been 8 years since I last used them.

level 4
3 points · 3 years ago

No no, I was saying that some students, quite literally, copy the solutions directly from Symbolab and it is extremely obvious. As in their solution will look identical to the solution listed there. It's analogous to a student plagiarizing material from a website and it sticks out like a sore thumb.

No, use of such software is not allowed on tests, and often even calculators are not allowed on tests, so students who have been relying on Wolfram and Symbolab to do their homework fail the tests because they have no idea how to actually solve the problems.

level 2
-6 points · 3 years ago(19 children)
level 3
22 points · 3 years ago

I don't know. My homework in college was with few to little points. It was basically just to practice for exams. If they are going to take exams, they'll still need to learn it.

level 4
9 points · 3 years ago

Being shown the steps is extremely important if they're going to apply them later. Many students struggle hard to go from A to Z, being able to go from A to B easily but not B to C, teachers have often been the way for a student to see how to go from B to C, but teachers sparse and busy, and often get fed up with students that struggle to make the logical jump from B to C even after several minutes of discussion. People learn differently, and sometimes need to review material over and over until it sinks in. I, for one, am all for websites that show step by step calculations. However, I think that these websites might be able to add a layer of interactivity that would add to the learning experience.

For instance, masteringphysics has that layer of interactivity, but I think it fails in that hints just don't often reveal enough to learn from. Yes, an apt student shouldn't really need more than the hint, but lets face it, a heavy course load, limited time, and the demand for higher grades make it so that struggling for, say, 2 hours, on a question because you can't go from B to C. Sometimes this struggle isn't a good thing, and leads to less motivated students.

level 3
10 points · 3 years ago

As someone on the front lines of rdycating university students, I can safely say generations of engineers will have no idea how to solve a basic problem because sites like these.

As someone on the frontlines of the engineering industry, I can safely say most engineers don't solve equations, let alone anything involving calculus.

level 3
7 points · 3 years ago

Engineers already use programs like Mathematica that do a lot of math for them. It's not lazy, it's efficient, and is a lot less prone to human error.

It's just a more advanced calculator. You still need to know how to use math to solve problems.

level 3
Number Theory3 points · 3 years ago

Despite the fact that everyone seems to believe the availability of software like this hurts student understanding, all the data we have tells us it helps students understand. Sometimes getting quick answers without getting bogged down in calculations lets students concentrate their learning on the parts that matter.

level 3
2 points · 3 years ago

Most of the students who'll use this instead trying themselves probably already copy solutions from others, bad students are going to be bad students.

level 4
1 point · 3 years ago

Or not bother with the assignment at all.

level 3
3 points · 3 years ago

Heh, probably one of the best things to happen to me in University was getting a "C" every time I retook pre-calc 2. I was never satisfied with my grade, so I kept taking the class over again. I slowly learned trig, though I really got good at it through lots of practice and hard work. That's the only way to get good at math, there is no substitute (at least, not until we are taken over by robot overlords).

level 4
1 point · 3 years ago

Why are you giving advice how to get good some at math when you got a C over and over in pre calc. Obviously what you are doing is not working.

level 5
2 points · 3 years ago

On the contrary: this person knows how to go from not being good at math to getting better (assuming they ever progressed past C) and probably knows literally every possible thing that does not work.

level 6
1 point · 3 years ago

If something significant changed in his approach the year he got a better grade then I would agree.

Else it is similar to someone crawling from one place to another and when he finally gets there he says "well, now I know how to get from one place to another, you just gotta keep crawling and finially you get there".

I wouldn't call him the best person to learn how to move from different places in a good way.

level 7
1 point · 3 years ago

You don't actually know that. All the GP said was he kept retaking the class and getting Cs. Maybe the last time he got the top 10 score ever, who knows?

level 8
1 point · 3 years ago

But then likely something drastically changed in his method last time.

level 5
1 point · 3 years ago
1. You're being an asshole.

2. Because I eventually got through it. I don't consider myself an expert at mathematics like most people here, but I do use it quite a bit at work and I'm decent at what I use. I'm quite comfortable with mathematics and feel like I can learn anything I need to because of my rigorous experience in school. What gets a person good at math is not taking a class over and over again, it is the side effect of rote practice.

level 6
0 points · 3 years ago
1. Hm, maybe. In these settings where you talk to a random person it is sometimes easier to not sugar coat things in 10 layers and instead just say what your point in one or two sentences. I expect the same to be done to me. Maybe that makes me an asshole.

2. I am not an expert at math either, it wasn't my point. My point was that you where stating your view on how to get good at math as a certain fact. Just because someone took a long time to achiveve something does in my view not imply that they are using a good method to achieve it. I would say that the contrary is more likely. If I would like to learn how to pass pre-calc would I ask for the study method from the one who passed it the first year or the one who passed it the fifth?

level 7
2 points · 3 years ago
1. Fair enough. It's hard to gauge tone and intent with internet text. I don't need anything sugar coated either, but that isn't a license to be an ass either. Anyways, I'm over it. Let's move on.

2. I may have made my point rather poorly. I read an article online a few months ago (and can't find it now) about how the best way to learn mathematics is still through rote memorization and lots and lots of practice. Using my own experience as anecdotal evidence, I can back up the claim of the article. Do a thousand trig problems correctly, and you'll eventually get the hang of it. It doesn't matter who you are. A professional mathematician has put in thousands of hours of practice, so they've developed a keen intuition for math.

I certainly deserve to be called out on my poor performance in the math class. I hadn't developed good study habits and this particular math class was the class which kicked my ass over and over again until I figured it out. Technically, I passed each time with a "C", but I needed to do better. Until then, all I had to do to get A's in my prior math classes was to show up, pay attention, and go through the motions. I never actually had to work hard to study up to this point. This particular pre-calc 2 class taught by this particular professor was hard. He was a strict teacher who didn't allow notes on tests, but you could use a 3x5 index card on the final. In other words, you have to know the material. You can't rely on calculators or external notes to help you -- they're crutches. The study habits I developed in that class is what allowed me to pass every subsequent "hard" class I took thereafter in university. This class wasn't the most difficult class I would take -- that award goes to a 400 level course of Elementary Logic -- but it would show me that there's nothing I can't figure out through working hard at it and being persistent.

level 3
1 point · 3 years ago · edited 3 years ago

These websites can benefit or hinder a student's learning habit. It is beneficial to find the solution in a short time and spend most of the time to understand it. My main concern is when students take the websites for granted. I feel like they look at the solution, believe they understand it, and move on. Then they will bomb the exam because they didn't actually understand. Since the website is free, more students will use it. Sometimes, students just need to do the math and struggle to actually understand and improve their math skills.

level 3
1 point · 3 years ago

There are engineers who have worked for more than a year who can still do calculus?

level 3
0 points · 3 years ago

As Albert Einstein I would agree. Here is a \$100.

level 1
1 point · 3 years ago

That's pretty awesome.

level 1
1 point · 3 years ago

I've used this to double check my work on random problems I made up to study for the finals coming up. Since I made them up I wanted to double check them. Stuff like this and wolframalpha are excellent for that. Of course, like others have said, I'm sure there will be students who abuse this in order to cheat on homework problems.

level 1
0 points · 3 years ago

Another pro tip is to learn how to use something like Mathematica. Not just "use it" for basic stuff, I mean like hard core learning how to use it. Set up functions, use lists, use the functional paradigms, etc. Bend it and break it until you can solve hundreds of homework problems in 5 seconds with it.

I guarantee that learning how to make a computer solve all of your homework problems, with all of the variations of the problems, will make you understand those equations in a way very few other things can do.

Sure, homework will take 5 times longer the first time around or whatever, but it'll take about 2 seconds the second time, and you'll understand it too.

level 2
Applied Math3 points · 3 years ago

For high school students, this might be quite the overkill.

level 3
0 points · 3 years ago

I started using LaTeX in high school. Overkill in high school turns into a lifesaver in college. I had so much free time in high school I had no idea what to do with myself. If I could've channeled my gaming addiction into something useful I could probably have professional level knowledge of programming by now, or be a lot stronger, or... lots of things.

Ah well. Now I'm just stuck here trying to pass principles of physics and failing semi-miserably, heh.

level 1
Comment deleted3 years ago(0 children)
level 2
15 points · 3 years ago

It's actually a really good resource and can help you understand difficult problems. It's one of those tools you use after spending an hour trying to figure it out yourself, and when you still can't understand Wolfram's solution after that. Symbolab will let you see the steps and it can be very elucidating.

level 2
11 points · 3 years ago

Why? Doing fancy integrals is a niche skill bordering on a parlor trick.

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text:
Determining the variables when training a/classifying with a random forest

Based on a response to the question "Statistical classification of text", I've decided that I want to use a random forest to perform various classifications of text.

Specifically, I am getting YouTube video data from their XML feeds. Here is an example:

<entry gd:etag="W/"DUQBR347eCp7I2A9WhdXFEQ."">
<published>2010-12-23T18:11:28.000Z</published>
<updated>2011-08-28T02:09:16.000Z</updated>
term="SSF4AE"/>
<!-- More categories/keywords here -->
<title>SSF4AE Mago 2D God VS Kazunoko aka Inoue</title>
<!-- More video information here, description, etc -->
</entry>

With that, I have very specific data that I believe I can use as the basis for variables in a random forest:

Variable    Value
--------    -----
Title       SSF4AE Mago 2D God VS Kazunoko aka Inoue
Category    SSF4AE

Looking at the video and based on the title/categories, I know that this is a video of the game Super Street Fighter 4: Arcade Edition (SSF4AE).

Looking at the alglib Decision Forest, I also read the general principals of data analysis methods, which speaks about encoding specifically for the algorithms on alglib; while I won't use it specifically, I believe it expanded my understanding of how data should be represented for a random forest.

Basically, using the above data, I'd need to transform each data point into a number of some sort.

For example, I could transform this into the following variables:

Variable                 Value
--------                 -----
Has "SSF4AE" in Title        1
Equals "SSF4AE"              1

Now, assuming I know that this video is for SSF4AE, and that I know that SSF4AE is a common acronym used to identify the game, would using the above as variables work for a random forest for training and classification?

Of course, that's a small set, and it's a very rough set of variables; my understanding is that a random forest basically will create nodes where there is a threshold for the variable, values over this fall on one side of the tree, values under on the other side.

In the above example, there are only two options, true or false.

However, I imagine I can do something like get the term-frequency/inverse document frequency and use that (using Lucene.NET or elasticsearch), like so (numbers are not accurate)

Variable                               Value
--------                               -----
Term Query for "SSF4AE" in Title        0.36
Term Query for "SSF4AE" in Category     1.00

And then train the random forest on that.

I assume that if the variables were done as above, then I could also place things such as boosts on the variables (in the tf/idf calculation) where I see that they are more prominent or I feel I want them to have more weight.

And finally, I can add more variables, but they have to be very specific. For example, I know that if there is a character Yun that only appears in SSF4AE, (not previous versions of Street Fighter 4), as well as in Street Fighter III. To that end, I can add two more variables:

Variable                               Value
--------                               -----
Term Query for "SSF4AE" in Title        0.36
Term Query for "SSF4AE" in Category     1.00
One character is Yun                       1
Term Query for "3S" in Title            0.00
Term Query for "3SO" in Title           0.00

("3SO" and "3S" being common acronyms for Street Fighter III: Third Strike)

However, for another video, which I know is for 3S, I the variables might look like this:

Variable                               Value
--------                               -----
Term Query for "SSF4AE" in Title        0.00
Term Query for "SSF4AE" in Category     0.00
One character is Yun                       1
Term Query for "3S" in Title            0.54
Term Query for "3SO" in Title           0.00

Where the training set would indicate that video is for 3S.

That said, are these feasible ways of going about determining the variables of a random forest?

Also, if I understand correctly, the random forest will also indicate which variables contributed the most to the classification, so I can discard them during classification later (if I want) or add different ones to get better classifications from my training data.

-
A few thoughts: (1) One measure of variable importance in RFs can be computed from a permuting scheme which is well described in any textbook or on Breiman's RF website. (2) It is safer to perform feature selection in a single shot (and within a proper cross-validation scheme), not using iterative search that will likely lead to overfitting; otherwise, consider using an hold-out sample. Sidenote: I imagine you already came across the following paper: Evaluation of Decision Forests on Text Categorization. – chl Sep 1 '11 at 19:34
@chl: I had not come across the paper, but thank you, it will definitely help. Admittedly, I am not a statistician and it took me about two weeks to get to the point I feel I could reasonably ask this question. That said, to your points 1) I just saw the section on permuting variables; it looks like it's not essential, just I won't be able to determine variable importance if I can't grok it (I have to understand out-of-bag better first). 2) I'm not sure what you mean by single shot, how to cross-validate, or what a hold-out sample would be. Could you elaborate, perhaps in an answer? – casperOne Sep 2 '11 at 16:43

As requested, I'll turn my comment to an answer, although I doubt it will provide you with a working solution.

The Random Forest™ algorithm is well described on Breiman's webpage. See references therein. Basically, it relies on the idea of bagging (bootstrap aggregation) where we "average" the results of one (or more) classifier run over different subsamples of the original dataset (1). This is part of what we called more generally ensemble learning. The advantage of this approach is that it allows to keep a "training" sample (approx. 2/3 when sampling with replacement) to build the classifier, and a "validation" sample to evaluate its performance; the latter is also known as the out-of-bag (OOB) sample. Classification and regression trees (CART, e.g. (2)) work generally well within this framework, but it is worth noting that in this particular case we do not want to prune the tree (to avoid overfitting; on the contrary, we deliberately choose to work with "weak" classifiers). By constructing several models and combining their individual estimates (most of the times, by a majority vote), we introduce some kind of variety, or fluctuations, in model outputs, while at the same time we circumvent the decision of a single weak classifier by averaging over a collection of such decisions. The ESLII textbook (3) has a full chapter devoted to such techniques.

However, in RF there is a second level of randomization, this time at the level of the variables: only random subsets of the original variables are considered when growing a tree, typically $\sqrt{p}$, where $p$ is the number of predictors. This means that RF can work with a lot more variables than other traditional methods, especially those where we expect to have more observations than variables. In addition, this technique will be less impacted by collinearity (correlation between the predictors). Finally, subsampling the predictors has for a consequence that the fitted values across trees are more independent.

As depicted in the next figure, let's consider a block of predictors $X$ ($n\times p$) and a response vector $Y$, of length $n$, which stands for the outcome of interest (it might be a binary or numerical variable).

Here, we'll be considering a classification task ($Y$ is categorical). The three main steps are as follows:

1. Take a random sample of the $n$ individuals; they will be used to build a single tree.
2. Select $k<p$ variables and find the first split; iterate until the full tree is grown (do not prune); drop the OOB data down the tree and record the outcome assigned to each observation.

Do steps 1--2 $b$ times (say, $b=500$), and each time count the number of times each individual from the OOB sample is classified into one category or the other. Then assign each case a category by using a majority vote over the $b$ trees. You have your predicted outcomes.

In order to assess variable importance, that is how much a variable contributes to classification accuracy, we could estimate the extent to which the predictions would be affected if there were no association between the outcome and the given variable. An obvious way to do this is to shuffle the individual labels (this follows from the principle of re-randomization), and compute the difference in predictive accuracy with the original model. This is step 3. The more predictive a variable is (when trying to predict a future outcome--this really is forecasting, because remember that we only use our OOB sample for assessing the predictive ability of our model), the more likely the classification accuracy will decrease when breaking the link between its observed values and the outcome.

• Reliable ways to assess the importance of your predictors have been proposed in the past (this might be the one described above, or additional resampling strategy related to backward elimination (4,5)); so there's no need to reinvent the wheel.
• The hard task, IMO, is to choose your descriptors or features, that is the characteristics of your records. Note that it should not be specific to a subset of your units, but rather encompasses a general description of possible events, e.g. "One Shot Game" vs. other (I'm not very good at game :-).
• Do an extensive literature search on what has been carried out in this direction on this specific field.
• Do not break the control on overfitting that is guaranteed by RF by running additional feature selection + new classification tasks on the same sample.

If you're interested in knowing more about RF, there's probably a lot of additional information on this site, and currently available implementations of RF have been listed here.

References

1. Sutton, C.D. (2005). Classification and Regression Trees, Bagging, and Boosting, in Handbook of Statistics, Vol. 24, pp. 303-329, Elsevier.
2. Moissen, G.G. (2008). Classification and Regression Trees. Ecological Informatics, pp. 582-588.
3. Hastie, T., Tibshirani, R., and Friedman, J. (2009). The Elements of Statistical Learning: Data Mining, Inference, and Prediction (2nd ed.). Springer.
4. Díaz-Uriarte, R., Alvarez de Andrés, S. (2006). Gene selection and classification of microarray data using Random Forest. BMC Bioinformatics, 7:3.
5. Díaz-Uriarte, R. (2007). GeneSrF and varSelRF: a web-based tool and R package for gene selection and classification using Random Forest. BMC Bioinformatics, 8:328.
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# How do you write the partial fraction decomposition of the rational expression  6/(x^2-25)^2?

Dec 30, 2015

$\frac{6}{{x}^{2} - 25} ^ 2$

$= - \frac{3}{250 \left(x - 5\right)} + \frac{3}{50 {\left(x - 5\right)}^{2}} + \frac{3}{250 \left(x + 5\right)} + \frac{3}{50 {\left(x + 5\right)}^{2}}$

#### Explanation:

Note that:

${\left({x}^{2} - 25\right)}^{2} = {\left(\left(x - 5\right) \left(x + 5\right)\right)}^{2}$

So need to solve:

$\frac{6}{{x}^{2} - 25} ^ 2 = \frac{A}{x - 5} + \frac{B}{x - 5} ^ 2 + \frac{C}{x + 5} + \frac{D}{x + 5} ^ 2$

$= \frac{A x + \left(B - 5 A\right)}{x - 5} ^ 2 + \frac{C x + \left(D + 5 C\right)}{x + 5} ^ 2$

$= \frac{\left(A x + \left(B - 5 A\right)\right) {\left(x + 5\right)}^{2} + \left(C x + \left(D + 5 C\right)\right) {\left(x - 5\right)}^{2}}{{x}^{2} - 25} ^ 2$

$= \frac{\left(A x + \left(B - 5 A\right)\right) \left({x}^{2} + 10 x + 25\right) + \left(C x + \left(D + 5 C\right)\right) \left({x}^{2} - 10 x + 25\right)}{{x}^{2} - 25} ^ 2$

$= \frac{\left(A + C\right) {x}^{3} + \left(B + D + 5 A - 5 C\right) {x}^{2} + \left(10 B - 10 D - 25 A - 25 C\right) x + 25 \left(B + D - 5 A + 5 C\right)}{{x}^{2} - 25} ^ 2$

Hence:

$A + C = 0$

$B + D + 5 A - 5 C = 0$

$10 B - 10 D - 25 A - 25 C = 0$

$25 \left(B + D - 5 A + 5 C\right) = 6$

From the first of these $C = - A$, so the rest become:

$B + D + 10 A = 0$

$10 \left(B - D\right) = 0$

$25 \left(B + D - 10 A\right) = 6$

From the second of these $D = B$, so the rest become:

$2 B + 10 A = 0$

$25 \left(2 B - 10 A\right) = 6$

From the first of these $B = - 5 A$, so the second becomes:

$25 \left(- 20 A\right) = 6$

Hence $A = - \frac{3}{250}$, $B = \frac{3}{50}$, $C = \frac{3}{250}$, $D = \frac{3}{50}$

So:

$\frac{6}{{x}^{2} - 25} ^ 2$

$= - \frac{3}{250 \left(x - 5\right)} + \frac{3}{50 {\left(x - 5\right)}^{2}} + \frac{3}{250 \left(x + 5\right)} + \frac{3}{50 {\left(x + 5\right)}^{2}}$


text:
Wednesday, January 23, 2008

Come on BBC, this is getting silly

Labels:

If you enjoyed this blog post, you might enjoy my travel book for people interested in science and technology: The Geek Atlas. Signed copies of The Geek Atlas are available.

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# prove or disprove: statement about Span

• Dec 25th 2012, 12:23 PM
Stormey
prove or disprove: statement about Span
Hi.

is this true or false:

$Span\left \{ u \right \}=Span\left \{ u,v \right \}\cap Span\left \{ u,w \right \}$

$u\neq 0$

BTW,
i think it is true, but i'm not sure...(Worried)
• Dec 25th 2012, 12:34 PM
emakarov
Re: prove or disprove: statement about Span
What if v = w?
• Dec 25th 2012, 01:00 PM
Stormey
Re: prove or disprove: statement about Span
(Angry)