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# Controlling symmetry and localization with an artificial gauge field in a disordered quantum system

## Abstract

Anderson localization, the absence of diffusion in disordered media, draws its origins from the destructive interference between multiple scattering paths. The localization properties of disordered systems are expected to be dramatically sensitive to their symmetries. So far, this question has been little explored experimentally. Here we investigate the realization of an artificial gauge field in a synthetic (temporal) dimension of a disordered, periodically driven quantum system. Tuning the strength of this gauge field allows us to control the parity–time symmetry properties of the system, which we probe through the experimental observation of three symmetry-sensitive signatures of localization. The first two are the coherent backscattering, marker of weak localization, and the recently predicted coherent forward scattering, genuine interferential signature of Anderson localization. The third is the direct measurement of the β(g) scaling function in two different symmetry classes, allowing to demonstrate its universality and the one-parameter scaling hypothesis.

## Introduction

Symmetry, disorder, and chaos are ubiquitous in both classical and quantum physics. These concepts are intimately intertwined: In a disordered crystal for instance, disorder stems from the absence of translational symmetry. But this does not mean that symmetries are absent in disordered/chaotic systems; on the contrary, they play a central role, as systems presenting the same symmetries display analogous properties. This idea led to the fundamental concept of universality class, grounding the famed random matrix theory1. In a dirty metal for instance, breaking the time-reversal symmetry (T-symmetry) has a profound effect on transport observables like electrical and thermal conductivities2. A common way to break the T-symmetry for charged particles is to apply a magnetic field. For neutral systems, where magnetic fields are inoperative, the concept of artificial gauge field3,4,5 has been introduced. It consists in building Hamiltonians that behave as if a gauge field were present. Artificial gauge fields have been realized in inhomogeneous or lattice systems, and very recently in the presence of disorder6.

In the present work, we exploit the simplicity and flexibility of driven cold-atom systems to generate such an artificial gauge field. For this purpose, we build on the well-known atomic kicked rotor7, a paradigm of both classical and quantum Hamiltonian chaos, which can be mapped onto an Anderson-like Hamiltonian in any dimension8,9. This system is realized experimentally by submitting laser-cooled atoms to short pulses (kicks) of a far-detuned laser standing wave.

By engineering the periodic driving, we obtain an experimental knob providing complete control of the relevant symmetry of the system, here the product of parity and time-reversal (PT-symmetry)10,11,12. Furthermore, the accumulated phase of a quantum particle along a closed multiple-scattering path is independent of the sense in which the loop is traveled when PT-invariance holds (defining the so-called orthogonal symmetry class), but not when it is broken (defining, for spinless systems, the unitary class), an effect that strongly affects quantum interference in localization phenomena. This allows us to directly observe the impact of this symmetry changing on interference signatures of localization in disordered media, and to study the universal transport properties in the two symmetry classes.

## Results

### Artificial gauge fields in disordered Floquet systems

We first show how to engineer the driving of Floquet systems to manipulate their fundamental symmetry properties. For this purpose, we consider a generalized kicked rotor Hamiltonian, to which we add a temporal dependence of the amplitude $${\cal K}(t)$$ and of the spatial phase a(t) in the potential term, both periodic in time:

$$\hat H = \frac{{\hat p^2}}{2} + {\cal K}(t)\,\,{\mathrm{cos}}\left[ {\hat x - a(t)} \right]\mathop {\sum}\limits_n {\kern 1pt} \delta (t - n),$$
(1)

The position $$\hat x$$ and momentum $$\hat p$$ are expressed in dimensionless units, and satisfy the canonical commutation relation $$[\hat x,\hat p] = i\hbar_{\mathrm{e}}$$, with $$\hbar_{\mathrm{e}}$$ playing the role of an effective Planck’s constant (see Methods for definition of units). When $${\cal K} = {\mathrm{const}}$$. and a = 0, we recover the standard kicked rotor, which can be mapped onto an Anderson-like tight-binding model in momentum space7,8 with diagonal pseudo-disorder.

When $${\cal K}(t)$$ is temporally modulated at a period 2π/ω2 incommensurate with the kick period, it has been shown9,13,14 that the temporal modulation can be taken into account by adding an effective spatial coordinate x2 = ω2t + φ along a synthetic dimension labeled “2” (“1” refers to the physical dimension along which all measurements are performed). Here, we study the situation where the driving modulations have a period which is an integer multiple of the kick period (ω2 = 2π/N), i.e. $${\cal K}(t + N) = {\cal K}(t)$$ and a(t + N) = a(t) with N an integer. In this case, the synthetic dimension is also periodic with twisted boundary conditions (see below). Such a system maps onto a synthetic nanotube threaded by an artificial gauge field. The flux of this artificial gauge field through the transverse section of the nanotube can be easily controlled by changing the initial phase φ of the temporal modulation.

Without loss of generality, it is convenient to illustrate the fundamental mechanism of creation and control of the artificial gauge field by using the specific example of a period-N amplitude modulation (see Fig. 1a, b):

$${\cal K}(t) = K\left[ {1 + {\mathrm{cos}}\left( {\frac{{2\pi t}}{N} + \varphi } \right)} \right]\,{\mathrm{and}}\,\,a(t) = 0.$$
(2)

The temporal dynamics of an arbitrary initial state χ(x) can be mapped on that of a two-dimensional pseudo-rotor with Hamiltonian14: $${\cal H} = p_1^2/2 + 2\pi p_2/N + K{\kern 1pt} {\mathrm{cos}}{\kern 1pt} x_1\left[ {1 + {\mathrm{cos}}{\kern 1pt} x_2} \right]$$$$\mathop{\sum}\nolimits_n {\kern 1pt} \delta (t - n)$$, with initial condition χ(x1)δ(x2 − φ). Here, “1” represents the physical dimension (x1 = x, p1 = p), and the synthetic dimension “2” is an ancillary space with 0 ≤ x2 < 2π, where the period-N dynamics is simply given by x2 = φ + 2πt/N (mod. 2π). This equivalent 2D Hamiltonian is time-periodic with period 1. Its Floquet states—eigenstates of the evolution operator over one period with eigenvalue eiω—are also solutions of a tight-binding model: $$\epsilon _{\bf{m}}{\mathrm{\Psi }}_{\bf{m}} + \mathop {\sum}\nolimits_{\bf{r}} {\kern 1pt} W_{\bf{r}}{\mathrm{\Psi }}_{{\bf{m}} - {\bf{r}}} = 0$$ where m ≡ (m1, m2) and r label the sites of a 2D square lattice which correspond to momenta in units of the effective Planck’s constant $$\hbar_{\mathrm{e}}$$, and Ψ m are the components of the Floquet quasi-states. The site energy $$\epsilon _{\bf{m}}$$ is $$\epsilon _{\bf{m}}$$ = $${\tan}\left\{ {\left[ {\omega - \left( {\hbar_{\mathrm{e}}m_1^2/2 + 2 \pi m_2/N} \right)} \right]/2} \right\}$$ and the hopping amplitudes W r are coefficients of the twofold Fourier expansion of $$W({x}_{1},{x}_{2})=\tan\left[K\cos{x}_{1}(1+\cos{x}_{2})/2\hbar_{\mathrm{e}}\right]$$9.

When $$\hbar_{\mathrm{e}}$$ is incommensurate with 2π, the site energies constitute a pseudo-random sequence in the direction “1”, which accounts for the disordered character of our system in momentum space, leading to dynamical localization, i.e., Anderson localization in momentum space. The site energies $$\epsilon _{m_1,m_2}$$ form a pseudo-random set which is periodic along the direction “2” with period N. Thus, we can use the Bloch theorem along the direction 2 and write any Floquet state as $${\mathrm{\Psi }}_{m_1,m_2} = {\mathrm{e}}^{ - {\mathrm{i}}m_2\phi _2}{\kern 1pt} \psi _{m_1,m_2}$$, where ϕ2 is the Bloch phase and $$\psi _{m_1,m_2 + N} = \psi _{m_1,m_2}$$ is periodic in direction 2. This is equivalent to having twisted boundary conditions along m2. The initial state $$\left| {\mathrm{\Xi }} \right\rangle$$ of the 2D pseudo-rotor—whose wavefunction is $$\left\langle {x_1,x_2{\mathrm{|\Xi }}} \right\rangle = \chi (x_1)\delta (x_2 - \varphi )$$ can be expanded in the momentum basis: $$\left| {\mathrm{\Xi }} \right\rangle = \mathop {\sum}\nolimits_{m_1,m_2} {\kern 1pt} {\mathrm{\Xi }}_{m_1,m_2}\left| {m_1,m_2} \right\rangle$$. The initial condition in direction 2 can be expanded in the momentum eigenbasis as: $$\delta (x_2 - \varphi ) = \mathop {\sum}\nolimits_{m_2 = - \infty }^{ + \infty } {\kern 1pt} {\mathrm{e}}^{ - {\mathrm{i}}m_2\varphi }{\mathrm{e}}^{{\mathrm{i}}m_2x_2}$$14 which implies $${\mathrm{\Xi}}_{m_1,m_2 + N} = {\mathrm{e}}^{ - {\mathrm{i}}N \varphi} {\mathrm{\Xi }}_{m_1,m_2 + N}$$ Thus, the choice of the initial condition imposes the Bloch phase ϕ2 = φ.

The Hilbert space thus reduces to a synthetic nanotube along direction 1, with N sites in the transverse section along direction 2 (Fig. 1c, d). The initial phase φ of the temporal modulation controls the flux through the nanotube. Indeed, the Floquet eigenequation can be rewritten for the periodic function ψ as:

$$\epsilon _{m_1,m_2}\psi _{m_1,m_2} + \mathop {\sum}\limits_{r_1,r_2} {\kern 1pt} W_{r_1,r_2}{\kern 1pt} {\mathrm{e}}^{{\mathrm{i}}\varphi r_2}{\kern 1pt} \psi _{m_1 - r_1,m_2 - r_2} = 0$$
(3)

The hopping matrix elements in Eq. (3) have caught a phase φr2. This is somewhat similar to a 2D system exposed to a uniform magnetic field15,16. However, the geometry here is not that of a planar system, but rather a quasi-1D system or a nanotube infinite along direction 1 and with N transverse sites along direction 2. Indeed, a closed loop m2 = 0 → 1 → 2… → N − 1 → 0 will pick a total phase Φ2 = , while the counter-propagating loop will pick the opposite phase −. In contrast, no phase is picked along a plaquette (m1, m2) → (m1 + 1, m2) → (m1 + 1, m2 + 1) → (m1, m2 + 1) → (m1, m2). Thus, the effective gauge field flux Φ2 is analogous to a magnetic flux, with the magnetic field along the axis “1” of the nanotube.

If the modulation period N is ≥3, a generic value of φ corresponds to a non-vanishing (mod. π) flux Φ2. In such a situation, it is not possible to unwind all the phases in Eq. (3) and the system is expected to be in the unitary symmetry class, where all anti-unitary symmetries—product of time-reversal by a geometrical unitary operation—are broken. (The case N = 2 is special, as the nanotube then degenerates in a two-leg ladder with a single transverse hopping matrix element, hence the system is in the orthogonal class whatever φ). In contrast, if Φ2 = 0 (mod. π), all hopping terms can be made real and the system is expected to be in the orthogonal class:

$${N \varphi = 0} \left( {{\mathrm{mod}} {.} {\pi}} \right) \,{:} \,{\mathrm{orthogonal}}{\mathrm{class}} \\ {N \varphi \ne 0} \left( {{\mathrm{mod}} {.} {\pi}} \right) \,{:} \, {\mathrm{unitary}}{\mathrm{class}}{.}$$
(4)

This simple condition can also be deduced from a direct analysis of the kick sequences (Fig. 1a, b). For the kicked rotor (1), the relevant anti-unitary symmetry is the product of time-reversal by parity (PT-symmetry)10,11. The Hamiltonian being explicitly time-dependent, there exists a family of operators $${\cal T}_\tau$$ : t → 2τ − t; x → −x; p → p, depending on the temporal origin of the time reversal. The condition for $${\cal T}_\tau$$ to be a symmetry operation requires that the sequence of kick amplitudes $${\cal K}(t)$$ be symmetric around some time τ (Fig. 1a). For the kick sequence in Eq. (2), it is easy to show that this happens only when φ is an integer multiple of π/N, in complete agreement with Eq. (4).

In the more general case of the Hamiltonian (1), it requires additionally that the kick phases a(t) be antisymmetric (as $${\cal T}_\tau$$ changes x to −x). Note that, in this general case, the mapping of the system on a nanotube is slightly more complicated, but the universality class is determined by the symmetric (resp. antisymmetric) character of the kick amplitudes (resp. phases). The Hamiltonian (5) used in the next section can be mapped on a bilayer nanotube with a magnetic field, and the Hamiltonian used in the last section on a nanotube with a more complicated artificial gauge field (the details of these mappings will be published elsewhere).

### Coherent back and forward scattering

Interference phenomena, which are at the core of Anderson localization, are very sensitive to symmetry breaking. Coherent backscattering (CBS)17 is a simple example: a consequence of the PT-symmetry is that pairs of scattering paths associated with the same geometrical loop, but traveled in opposite senses, accumulate the same quantum phase and thus interfere constructively. When the symmetry is broken, these pairs of paths become out of phase and CBS disappears. However, in the presence of (strong) Anderson localization, other non-trivial quantum interference effects exist, such as the coherent forward scattering (CFS), recently predicted theoretically18 (see also ref. 19 in the context of the kicked rotor). Contrary to CBS, the CFS is insensitive to the symmetry breaking and, for unbound systems, requires the onset of Anderson localization in order to show up20,21,22,23. While experimental observations of CBS have been achieved in many different systems24,25,26,27,28,29, to the best of our knowledge no experimental observation of the CFS had been reported up till now.

In spatially disordered systems, CBS and CFS manifest in the reciprocal space as two peaks centered around −k0 (backward) and +k0 (forward) directions of the velocity distribution of a wave packet initially launched with a well-defined wave vector k018. Alternatively, the constructive interference between time-reversed loops also manifests in the direct (configuration) space by an enhanced probability to return to the original position17.

This interference is visible, in our system, in a mixed momentum/configuration space representation (p1,x2), in which the initial state is localized. Starting with the initial conditions p1(t = 0) ≈ 0 and x2(t = 0) = +φ, a CBS peak should be observed around p1 = 0 at x2 = −φ (in the presence of the PT-symmetry) and a CFS peak around p1 = 0 at x2 = +φ19. Because of the time-dependence of x2(t) = x2(0) + 2πt/N, we thus expect to observe CBS and CFS at different times, depending on the initial phase φ of the modulation (see Methods). Both CBS and CFS are measurable in the physical dimension p1 as peaks around the initial momentum p1 ≈ 0. The temporal modulation is thus essential to separate them, so that they appear at different moments during the kick sequence.

We experimentally studied the CBS and CFS effects by using a thermal, ultra-cold cloud of Cs atoms kicked by a series of short pulses of a far-detuned standing wave (see Methods) created by a pair of counter-propagating laser beams. The amplitude and relative phase of the beams can be changed from one kick to another in order to create arbitrary sequences $${\cal K}(t)$$ and a(t). After the application of the kicks, we perform a time-of-flight measurement of the probability density in momentum space $${\mathrm{{\Pi}}}(p_1,t) = \left| {{\mathrm{\Psi }}(p_1,t)} \right|^2$$ vs. time t. This gives us access to the temporal dynamics of the return probability17, i.e. the zero-momentum probability density $${\mathrm{{\Pi}}}_0(t) = \left| {{\mathrm{\Psi }}(p_1 = 0,t)} \right|^2$$. Using this observable, we are able to distinctly observe the CBS and CFS effects, and to study their dynamics and behavior with respect to the relevant symmetries of the system.

Although it is possible to observe CBS and CFS using the Hamiltonian (2) discussed above, it turns out (Methods) that the two peaks can be better separated temporally by using a combined modulation of amplitude and phase. For this reason, in our experiment we utilize the following period-10 Hamiltonian, which also belongs to the class of models (1):

$${\cal K}(t) = K\left[ {1 + {\mathrm{cos}}\left( {\frac{{2\pi (t - 1)}}{5}} \right)} \right]\,{\mathrm{and}}\,\, a(t) = - a,t\, {\mathrm{even}}\\ {\cal K}(t) = K\left[ {1 + {\mathrm{cos}}\left( {\frac{{2\pi (t - 1)}}{5} + \tilde \varphi } \right)} \right]\,{\mathrm{and}}\,\,a(t) = a,t\, {\mathrm{odd}}{{.}}$$
(5)

As explained above, the symmetry properties of the Hamiltonian (5) are controlled by tuning the parameter $$\tilde \varphi$$, while the additional phase modulation a(t), with period 2, makes CBS and CFS observable only at even kicks. The CFS peak is observed at kicks multiples of the period N = 10 of the system: 10, 20, 30… (see Methods). The CBS peaks, on the other hand, can exist only if the Hamiltonian is PT-symmetric. This is the case if $$\tilde \varphi$$ is an integer multiple of 2π/5, when the kick sequence displays PT-symmetry axes τ, as illustrated in Fig. 2a. The CBS is then observed at kicks which are symmetric to the initial kick with respect to these axes, that is at times 6, 16, 26… (see Methods). In contrast, when $$\tilde \varphi$$ is not multiple of 2π/5 (see Fig. 2b) the sequence has no symmetry axis, and no CBS shows up. We thus see that a suitable choice of the symmetry properties allows for a clear temporal separation of the two peaks.

The experimental results for $$\tilde \varphi = 0$$ (Fig. 2c) display two characteristic features: First, the general trend of Π0(t) is a decay vs. t, due to the spreading of the initially narrow wave packet in momentum space. This decay slows down at long times, when localization sets in. Second, we observe pronounced peaks at kicks 20, 26, 30, 36, etc. From this series of peaks, one can however distinguish two subsequences with different properties: The CBS series at t = 6 (mod. 10) has a maximal contrast at the beginning, which slowly decreases with time, while the contrast of the CFS series at t = 0 (mod. 10) increases at short times. On a longer time scale (set by the localization time tloc), the CFS amplitude asymptotically converges toward the CBS one, and the two peaks become twins after localization has set in. This constitutes a direct interferential proof of the occurrence of Anderson localization. Adding a phase $$\tilde \varphi$$ to the modulation should break the PT-symmetry, as explained above. Indeed, this manifests (Fig. 2d) in the disappearance of the CBS peaks at t = 6 (mod. 10), whereas at pulses t = 0 (mod. 10) the CFS peaks survive and follow the increasing trend, until saturating at t ~ tloc.

To test their dependence on $$\tilde \varphi$$, we monitor the contrasts of the CBS and CFS peaks (see the Methods section for details on contrast definition and measurement procedure). The results are shown in Fig. 3a: at $$\tilde \varphi = - 2\pi /5$$ (which preserves the PT-symmetry) we observe a pronounced maximum of contrast for the CBS peaks, present here at kicks 2 (mod. 10) (see Methods). The decrease of the CBS contrast around this value is a clear signature of the symmetry breaking (an analogous control, albeit not breaking the symmetry, was developed in refs. 30,31 for CBS in spatially disordered ultracold experiments). Although the geometry is not strictly identical, this is qualitatively similar to the magneto-resistance effect32 induced in a solid-state sample when time-reversal symmetry is broken by an external magnetic field. On the other hand, the contrast of the CFS peak is insensitive to $$\tilde \varphi$$, showing its robustness vs. the PT-symmetry breaking. These effects are well reproduced by numerical simulations of wave packet evolution (solid lines in Fig. 3a) based on a Fourier transform method33.

The time-dynamics of the CFS contrast has been theoretically predicted in ref. 22, using a non-perturbative, fully time-resolved analytical description of a quantum quench in an Anderson-localized unitary system. Unlike the CBS peak, which is present at short times with maximal contrast, the CFS peak requires (strong) Anderson localization in order to show up, on a time scale set by the localization time tloc.

In our experiment (Fig. 3b), the slow decay of both peaks at longer times is due to stray decoherence. The CBS contrast follows an exponential decay C B (t) = C0 exp(−t/tdec)34 and is an excellent benchmark for decoherence in our system. A fit gives the decoherence time tdec ≈ 190 and the initial amplitude of the CBS contrast C0 ≈ 0.45 (lower than unity, due to a finite initial momentum width effect). In the unitary class, the CFS dynamics is very well fitted by the analytical formula of ref. 22 multiplied by the same exponential decay due to decoherence: C F (t) = C0I0(2tloc/t)exp(−2tloc/t)exp(−t/tdec) with tloc ≈ 40 the only fitting parameter (I ν is the modified Bessel function of order ν). Very recently, an analytical formula for the CFS dynamics in the orthogonal class has been given in ref. 35, which, when including decoherence effects, gives: C F (t) = C0[I0(2tloc/t) + I1(2tloc/t)]exp(−2tloc/t)exp(−t/tdec). A one-parameter fit using this expression is found to be in very good agreement with our experimental data, and gives tloc ≈ 37. Our measurements clearly confirm that the CFS contrast dynamics is faster in the Orthogonal class, because of the presence of simple loops favoring Anderson localization on a shorter time scale.

These observations demonstrate that the CFS is a marker of the non-trivial quantum interferences needed to build Anderson localization in quantum disordered systems. The fact that we can observe a destruction of CBS in the presence of a surviving CFS is a clear-cut proof of the PT-symmetry breaking, and that other effects, such as decoherence, are not at stake for the destruction of the CBS (Methods). Hence, this represents an unambiguous evidence of the changing of our system from the orthogonal to the unitary class under the effect of the artificial gauge field.

### Symmetry and transport—universal one-parameter scaling law

The interference phenomena leading to Anderson localization also dramatically influence the bulk transport in disordered quantum systems. First corrections to the classical (incoherent) diffusion coefficient D0, known as weak localization, come from CBS-type interferences which enhance the return probability of a quantum particle17. This quantum corrections are directly linked to the presence of the PT-symmetry. In absence of this symmetry, more complex CFS-type interferences induce a slower deviation from diffusive behavior, with a distinct form.

An instrumental progress in the theory of metal-insulator transitions was the so-called one-parameter scaling theory introduced by Abrahams et al.36. It shows that, irrespective of the microscopic details of the system, transport properties should obey a universal scaling behavior, characterized by a single quantity, $$\beta\equiv {\mathrm {d}}\ln (g)/{\mathrm{d}}\ln (L)$$, the logarithmic derivative of the dimensionless conductivity g with respect to the size L of the system, which is a measure of transport. Expressed only as a function of the conductivity g itself, the resulting β(g) function is universal, that is, independent of microscopic details. This function has played a central role in the study of disordered systems. Based on the momentum spreading of a kicked wave packet, we here present a direct and simple experimental measurement of the β(g) scaling function, for both orthogonal and unitary classes, as well as a test of its universality within each symmetry class.

As it maps on a pseudo-random Anderson model (see above), the kicked rotor should obey a one-parameter scaling law. It is however a dynamical system, so that one has to build dynamical quantities—which are the equivalent of the system size L and the dimensionless conductance g. The natural choice for the system size L is the number of lattice sites effectively populated. In momentum space, the lattice sites are momentum eigenstates separated by $$\Delta p=\hbar_{\mathrm{e}}$$, so that we can define $$L = \sqrt {\left\langle {p_1^2}(t)\right\rangle}/\hbar_{\mathrm{e}}$$. Following ref.37, we also define $$g = N\sqrt {\left\langle {p_1^2(t)} \right\rangle } /t \hbar_{\mathrm{e}}$$ as the effective conductance. In the classical regime where the dynamics is diffusive, this leads to $$g=2N D_{0}/\hbar_{\mathrm{e}} L$$, a perfectly reasonable result, as the conductance decreases like the inverse of the system size (Ohm’s law) and is proportional to the diffusion coefficient $$D_0$$ (Einstein’s law). The prefactor N takes into account the fact that our synthetic quasi-1D system consists of N transverse channels (see above), so that its conductance is N times larger than for a purely 1D system. Note that this definition immediately leads to β = −1, as expected for a classical diffusive quasi-1D system.

As a logarithmic derivative, the β(g) function is extremely sensitive to experimental noise. A proper estimation of β(g) also requires to average over a sufficiently large number of disorder realizations, chosen accordingly with the desired PT-symmetry properties defining the universality class. Achieving this goal is not possible with the Hamiltonian (5), which does not provide enough degrees of freedom. In the frame of Hamiltonian (1), the best strategy turns out to be an average over a periodic series of N randomly chosen phases $${\cal A}_N = \left\{ {a_1,a_2, \ldots ,a_N} \right\}$$ i.i.d. in [0, 2π], with $${\cal K}(t) = K = {\mathrm{const}}$$. Then, the $${\cal A}_N$$ series is PT-symmetric if a k  = −aNk+1, k integer (1 ≤ k ≤ N).

Using this prescription, we measured $$\left\langle {p_1^2(t)} \right\rangle$$ by averaging the experimental momentum distributions over a large number (100) of realizations of these random phases, with the microscopic parameters K, N, and $$\hbar_{\mathrm{e}}$$ fixed. In the absence of quantum interference, $$\left\langle {p_1^2} \right\rangle$$ evolves diffusively with time: $$\left\langle {p_1^2} \right\rangle = 2D_0t$$. In the orthogonal class, self-intersecting single-loop (CBS-like) interference paths, which are already present from very short times, lead to a rapid deviation from classical diffusion (Fig. 4a). In the unitary class, where the one-loop corrections are absent, this has a dramatic effect on transport properties, leading to a slower deviation from classical diffusion (Fig. 4b).

The leading corrections to $$\left\langle {p_1^2(t)} \right\rangle$$ due to loops have been calculated for the kicked rotor in ref. 38, both in the orthogonal and unitary classes (the unitary-class correction is given with the wrong sign in ref. 38—C. Tian, private communication). They allow us to compute the lowest-order correction to the β(g) function, valid in the limit of large conductivities:

$$\begin{array}{*{20}{l}} {\beta (g) = - 1 - \frac{{4\sqrt 2 }}{{3\sqrt \pi g}}} \hfill & : \hfill & {{\mathrm{orthogonal}}{\kern 1pt} {\mathrm{class,}}} \hfill \\ {\beta (g) = - 1 - \frac{1}{{2g^2}}} \hfill & : \hfill & {{\mathrm{unitary}}{\kern 1pt} {\mathrm{class}}{\mathrm{.}}} \hfill \end{array}$$
(6)

In order to test these predictions, and the universality of β(g), we studied a series of different values for the microscopic parameters K and N, in the two symmetry classes. The measured β-functions are shown in Fig. 4c. A remarkable feature of these results is that all data collapse on two distinct scaling functions, as evidenced by the shaded zones, characteristic for each universality class. This constitutes an experimental demonstration of the validity of the one-parameter scaling law. It also shows that the shape of the β(g) function constitutes a clear marker of the presence or absence of the PT-symmetry.

The experimental results in the unitary class are in excellent agreement with (6). This is also true in the orthogonal class, in the limit of large g (typically for 1/g < 0.5). For smaller values of g, we notice deviations from (6), which we confirmed through numerical simulations. This probably indicates that higher-order interference diagrams should be taken into account in the orthogonal class. These observations will hopefully stimulate further theoretical investigations in this direction.

## Discussion

The striking observations reported in the present work highlight the importance of symmetries for the localization and transport properties of disordered media, and the possibility to control them using an artificial gauge field—generated here by appropriately tailoring the driving parameters of a Floquet system. Our method presents a remarkable experimental simplicity, and avoids both the complexity and limitations in more involved schemes (using, e.g., close-to-resonance Raman-dressing of internal states). We characterized the Anderson localization from a different perspective, by directly probing interferential building blocks such as the coherent back- and forward-scattering phenomena. We also measured, in perfectly controlled conditions, the β(g) scaling function—a universal characteristic measure of transport in disordered media. Moreover, we demonstrated the different sensitivity of these effects with respect to the artificial gauge field flux, which controls the PT-symmetry properties of the system.

Interference signatures (such as the CFS) could provide valuable tools to observe the Anderson transition and probe its critical properties in higher dimensions and different symmetry classes. Engineering spin-orbit-coupled dynamical Floquet systems (e.g., using internal-state-dependent optical potentials) would allow, for example, to study the symplectic symmetry class, where the Anderson metal-insulator transition is expected to occur in dimensions as low as two. This also opens an avenue for the study of fascinating phenomena, like quantum Hall effect39, Floquet topological insulators and artificial magnetism.

## Methods

### Experiment

In the experiment, we start from a laser-cooled Cesium atomic sample, prepared in a thermal state ($$T \simeq 1.5$$ μK). The cloud is kicked along the vertical x axis by a far-detuned, pulsed (period T1) optical standing wave (SW), which is created by two independent lasers beams. This allows us to control the amplitude and phase of the potential (via the RF signal sent to two different AOMs) and to shape the modulation sequences $${\cal K}(t)$$ and a(t) as in Eq. (1). The laser parameters are: the detuning Δ = −13 GHz (at the Cs D2 line, wavelength λ = 852.2 nm), the waist w0 = 800 μm and the maximum intensity I = 30 W cm−2 per beam. The pulse duration is τ = 200 ns, while T1 is typically varied between 10 and 30 μs. After the desired number of kicks, the cloud is allowed to expand and the momentum probability density $${\mathrm{{\Pi}}}(p,t) = \left| {{\mathrm{\Psi }}(p,t)} \right|^2$$ is measured using a time-of-flight (TOF) technique34.

The TOF expansion time t = 170 ms corresponds to a dropping distance of $$g t^2/2=14\,{\mathrm {cm}}$$ along the vertical z direction. The atomic cloud is detected at this location using absorption imaging in a homodyne detection scheme. We utilize a 10 mW probe beam with corresponding waists w z  = 300 μm and w x  = 3.3 mm in the vertical and horizontal directions, respectively. The probe passes through a 45 MHz phase modulator, one of the sidebands being resonant with the atomic transition. The transmitted power of the probe is then detected using a fast photodiode, whose signal is demodulated in real-time using a spectrum analyzer. Residual magnetic field effects are negligible, as the currents in the MOT coils are switched-off at the beginning of the molasses phase, typically 50 ms before starting the kick sequence.

For the CBS/CFS measurements (Section “Coherent Back and Forward Scattering”), it is crucial to utilize a sample with an initial momentum distribution narrower than the width of a Brillouin zone. Indeed, the CBS and CFS peaks have widths given by that of the initial state, and their respective contrasts (equal to one in the ideal case) is strongly reduced otherwise. In order to decrease the mean kinetic energy of the sample, the atoms are loaded in a very shallow 1D optical lattice (vertical direction), whose depth is less than the initial temperature. This filters out the most energetic atoms. Subsequently, we realize 1D adiabatic cooling by switching off the lattice in ~ 1 μs, reaching a momentum distribution width <0.67 × 2ħk L , which corresponds to an equivalent 1D temperature <400 nK (this value is limited by the resolution of the time-of-flight detection). The filtering technique used for studying the CBS and CFS peaks is less suitable for the β(g) measurements, where starting from a narrow momentum distribution is less crucial whereas having a larger atom number is important to increase the detection signal-to-noise ratio. We thereby used a standard molasses (1.5 μK) as a starting point for these experiments (Section “Symmetry and transport: universal one-parameter scaling law”).

### Units

We have chosen scaled variables in order to express the Hamiltonian in the dimensionless form Eq. (1): distances along the x axis are measured in units of (2k L )−1 (where k L is the SW wave number), time in number of kicks (or units of T1), the particle mass is unity. The Hamiltonian (1) is associated with the Schrödinger equation $$i \hbar_{\mathrm{e}}{\partial \psi }/{\partial t} = \hat H\psi$$, where $$\hbar_{\mathrm{e}} \equiv 4\hbar k_L^2T_1/M$$ plays the role of an effective Planck constant, which can be adjusted at will by modifying, e.g., the kick period T1 (M is the Cs atomic mass). The canonical commutation relation reads $$[\hat x,\hat p] = i\hbar_{\mathrm{e}}$$.

The Hamiltonian (1) is spatially 2π-periodic, so that the solutions of the Schrödinger equation can always be expanded on a discrete lattice in momentum space $$p_{m}=(m+\beta)\hbar_{\mathrm{e}}$$, where m is an integer and −1/2 < β ≤ 1/2 is the quasimomentum, varying in the first Brillouin zone. Due to the spatial periodicity of the system, β is a constant of motion, so that the whole analysis can be performed for β = 0.

### Decoherence and choice of experimental Hamiltonians

Decoherence in our setup comes mainly from residual spontaneous emission and fluctuations in the SW phase, and is one of the major limitations of all the experiments presented here. To keep it under control, we use rather small average values of K, which somewhat conflicts both with the observation of the CBS and CBS peaks and with precisely measuring the β(g) scaling function, as explained below.

CBS/CFS peaks from Hamiltonian (5) In the Section corresponding to the CBS/CFS measurements, we use a relatively large (N = 10) modulation period, which is suitable for achieving a proper temporal separation between the CBS and CFS peaks. Moreover, in order to properly resolve the CFS dynamics, one also needs a sufficiently large tloc. For the kicked rotor this is usually obtained by increasing the kick amplitude K, which unfortunately decreases the decoherence time tdec in the experiment. On the other hand, it turns out that adding a period-two phase modulation increases (for certain values of the phase-shift a) the diffusion coefficient D0, and thus tloc, without affecting tdec. This is why, for optimizing the experimental conditions for the measurements of the CBS and CFS dynamics, we opted for the combination of phase and amplitude modulations (5), choosing the value a = 0.21 × π, found to maximize D0.

### β(g) measurements

The scaling function $$\beta\equiv {\mathrm{d}}\ln (g)/{\mathrm{d}}\ln (L)$$ is extremely sensitive to spurious effects such as experimental noise. In particular, at low values of the kick amplitude K (used to keep decoherence under control), short-time correlations between kicks are known to occur, and lead for instance to well-known oscillations in the diffusion coefficient13,40). These temporal correlations are responsible for large-amplitude oscillations of $$\left\langle {p^2(t)} \right\rangle$$, which are magnified when computing the β(g) function (defined as a logarithmic derivative).

In order to eliminate these effects and properly measure β(g), it is therefore required to average over as many disorder realizations as possible. For this purpose, the best strategy, as confirmed by numerical simulations, is to average over a large number of realizations of the random phase sequence a(t). This method was used for our β(g) measurements: Each experiment is repeated 500 times, with a total of 100 different random realizations of a(t), and resulting momentum distributions $$\left| {{\mathrm{\Psi }}(p,t)} \right|^2$$ are averaged. To determine $$\left\langle {p^2(t)} \right\rangle$$, we fit the measured distribution of squared-momenta $$\left| {p{\mathrm{\Psi }}(p,t)} \right|^2$$ using the Lobkis-Weaver expression of the momentum distribution41.

### Symmetry and times of occurrence of CBS and CFS peaks

The pulse sequence is modulated using a combination of amplitude and phase modulations, as in (1). The kick amplitude sequence $${\cal K}(t)$$ has a period of 5, whereas the phase a(t) is modulated with a period of 2 (represented in Fig. 4 by the different colors used for the even and odd kicks), with an overall period N = 10. A consequence of the period-two phase modulation a(t) is that PT-symmetry axes only occur in-between kicks (and never during a kick) which explains why CBS peaks do not occur for odd values of the kick number. A simple analysis of (5) shows that the corresponding Hamiltonian is PT-symmetric (belonging thus to the orthogonal class) when the phase $$\tilde \varphi \in 2\pi \times \left\{ {0,\frac{1}{5},\frac{2}{5},\frac{3}{5},\frac{4}{5}} \right\}$$. Each of these values of the $$\tilde \varphi$$ leads to a different time of occurrence of the CBS peak—corresponding to kicks {6, 10, 4, 8, 2}, respectively.

Take for instance the modulation sequence shown in Fig. 5, corresponding to experimental data in Fig. 2. When $$\tilde \varphi = 0$$ (Fig. 5a), the sequence has PT-symmetry axes (vertical dashed lines labeled, τ), and the system belongs to the orthogonal class. In this case, CBS peaks are expected to appear periodically, at kicks 6 (mod. 10), i.e., at times equal to twice the occurrence time of each τ. On the other hand, when $$\tilde \varphi \, \notin \, 2\pi \times \left\{ {0,\frac{1}{5},\frac{2}{5},\frac{3}{5},\frac{4}{5}} \right\}$$ no symmetry axes exist (e.g. in (Fig. 5b), for $$\tilde \varphi = - 3\pi /5$$). In both universality classes the symmetry-insensitive CFS peaks occur at integer multiples of the period of the system, i.e., at kicks 0 (mod. 10).

### The CBS and CFS contrast measurements

Analyzing the experimental data in Fig. 2, we can extract the contrasts C B (t) and C F (t), of the CBS and CFS peaks respectively, vs. time. The contrasts, for either case, are defined as: CB,F(t) = (Π0(t) − Π0,incoh.(t))/Π0,incoh.(t), and are evaluated at the occurrence times of their respective peaks, tCBS and tCFS (corresponding respectively to red and and green points in Fig. 2). Here, $${\mathrm{{\Pi}}}_0(t) = \left| {{\mathrm{\Psi }}(p_1 = 0,t)} \right|^2$$ is the total zero-momentum probability density (also defined in the main text), while Π0,incoh.(t) corresponds to the incoherent (classical) contribution to Π0(t). Outside tCBS and tCFS (i.e., at times corresponding to the black points in Fig. 2), the two contributions are identical: Π0(t) = Π0,incoh.(t). In order to evaluate CB,F(t), we interpolate the Π0,incoh.(t) values at tCBS and tCFS. This method was used for the data shown in Fig. 3.

### Data availability

The data that support the findings of this study are available from the corresponding author upon request.

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

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## Acknowledgements

We are grateful to Chushun Tian and Adam Rançon for fruitful discussions and to Antoine Browaeys for a critical reading of this manuscript. G.L. acknowledges an invited professorship at Sapienza University of Rome. G.L. thanks CalMiP for access to its supercomputer. This work is supported by Agence Nationale de la Recherche (Grant K-BEC No. ANR-13-BS04-0001-01), the Labex CEMPI (Grant No. ANR-11-LABX-0007-01), Programme Investissements d’Avenir under the program ANR-11-IDEX-0002-02, reference ANR-10-LABX-0037-NEXT, and the Ministry of Higher Education and Research, Hauts de France Council and European Regional Development Fund (ERDF) through the Contrat de Projets Etat-Region (CPER Photonics for Society, P4S).

## Author information

### Affiliations

1. #### CNRS, UMR 8523, Laboratoire de Physique des Lasers Atomes et Molécules, Université de Lille, 59000, Lille, France

• Clément Hainaut
• , Isam Manai
• , Jean-François Clément
• , Jean Claude Garreau
• , Pascal Szriftgiser
2. #### Laboratoire de Physique Théorique, IRSAMC, Université de Toulouse, CNRS, 31062, Toulouse, France

• Gabriel Lemarié
3. #### Department of Physics, Sapienza University of Rome, P.le A. Moro 2, 00185, Rome, Italy

• Gabriel Lemarié
4. #### Laboratoire Kastler Brossel, UPMC-Sorbonne Universités, CNRS, ENS-PSL Research University, Collège de France, 4 Place Jussieu, 75005, Paris, France

• Nicolas Cherroret
•  & Dominique Delande

### Contributions

C.H., I.M., J.-F.C., J.C.G., P.S., and R.C. performed the experiments and G.L., N.C., and D.D. performed the theoretical work. All authors contributed to the preparation of this manuscript.

### Competing interests

The authors declare no competing interests.


text:
# Function with oscillating frequency?

I'm looking for a function whose frequency oscillates around a certain value (say, oscillating between 440 Hz and 880 Hz, at a rate of 1 Hz -- i.e., its frequency goes up and down once per second, preferably in a somewhat linear fashion).

I feel like it should be quite simple, but I'm having trouble coming up with such a function.

I tried $x(t)=\sin(\sin(t)\ t)$, but this doesn't seem to work, because it's not periodic -- I'm looking for something whose frequency goes up and down in a periodic fashion.

Ideas?

-
When you ask two such closely related questions, it makes sense to cross-link them so that people don't waste their time answering overlapping parts of them separately. – joriki Nov 24 '11 at 22:55

As a simple model, something like $t\mapsto A\sin\big(\omega_1( t + \alpha\sin(\omega_2 t))\big)$ should do the trick, where
• $\omega_1$ is the (angular) frequency of the carrier wave.
• $\omega_2$ is the (angular) frequency of the signal.
• $\alpha$ is a modulation depth that should be kept comfortably below $1$.
If the signal is something more complex than a sine wave, replace $\sin(\omega_2 t)$ by an antiderivative of the signal.

format: plain


text:
# On a nonlinear wave equation

I am considering the following wave equation (for $$\phi=\phi(x,t)$$) $$\phi_{tt} - \Delta \phi = a |\nabla \phi|^2, \quad (x,t) \in \mathbb{R}^3 \times \mathbb{R}$$ where $$\nabla$$ is just spatial gradient, i.e., $$\nabla \phi= (\partial_{x_1} \phi, \partial_{x_2} \phi, \partial_{x_3} \phi)$$ and $$a \in \mathbb{R}$$. And assume that initial data $$\phi(x,0)$$ and $$\partial_t \phi(x,0)$$ are regular enough and compactly supported.

This PDE in some sense is the complementary case of the PDE in John's blow-up theorem, where we have $$\phi_{tt} - \Delta \phi = (\partial_t\phi)^2.$$

Now, I have the following questions: is this PDE well-posed? Or the solution will blow-up? Is there any paper on this PDE in 3+1 dimensions?

By replacing $$\phi$$ by $$-\phi$$ you can set $$\alpha$$ to be positive (or negative, if you wish). By replacing $$\phi$$ by $$\lambda \phi$$ you can rescale away $$\alpha$$. So you can set $$\alpha$$ to be either $$+1$$ or $$-1$$ as you wish.

So for now let's consider the situation you are solving $$\partial^2_{tt}\phi - \Delta \phi = |\nabla \phi|^2$$ First observation: suppose $$\phi$$ is a $$C^3$$ solution with compactly supported initial data, then $$\phi$$ is uniformly bounded from below on $$\mathbb{R}^3\times [0,\infty)$$. This is due to the fact that on $$\mathbb{R}^3$$ the fundamental solution to the wave equation is signed (a fact also used by John in his proof), so that the solution to the above equation is pointwise $$\geq$$ the solution to the linear equation with the same initial data. The latter is globally uniformly bounded, and hence $$\phi$$ is uniformly bounded below.

Now $$v = e^\phi - 1$$; you have that $$v$$ is also $$C^3$$, and there exists $$\beta$$ such that $$v + 1 > \beta$$ uniformly on $$\mathbb{R}^3\times [0\infty)$$, using the uniform bound on $$\phi$$.

Now we compute $$(\partial^2_{tt} - \Delta) v = e^\phi (\partial^2_{tt} - \Delta) \phi + e^\phi (\partial^2_t \phi - |\nabla\phi|^2) = \frac{1}{1+v}( \partial_t v)^2$$

The lower bound above ensures that the denominator on the RHS is never zero.

Suppose now that $$\phi$$ is globally bounded above also: then $$(1+v)$$ is bounded above globally by some $$M > 0$$. Then the condition for Theorem 2 in John's 1981 paper is satisfied and the result follows.

So the remaining question is whether it is possible for $$\phi$$ to blow-up only at infinity and not at any finite time. I think there may be a way to rule this out, but that would require checking some more technical details so I'll have to report back later.

• @Immanuel: all typos are now fixed (which should answer all of your three comments above). I need to think more about how (and whether we can) rule out blow-up at infinity. Apr 13 at 22:15
• local well-posedness is immediate. See e.g. Sogge's book. Of interest is small data global well-posedness. // Hyperbolicity only concerns the principal part of the equation. Your equation is semilinear so the principal part is just the standard linear wave operator, so you have hyperbolicity. You can run into problems if your nonlinearity depends on the second derivative of $\phi$, as in this case the principal symbol may be changed. Apr 13 at 22:20
• Yes, in spatial dimension $\leq 3$. See e.g. my paper arxiv.org/abs/1407.6276v1 for a survey of known results as of 8 years ago. The paper was written with focus on the quasilinear case, as that is where the blow-up results are more robustly proven. For the case of $\Box \phi = g(\partial\phi)$, the small data global well posedness results are basically the same as that for the quasilinear case. Apr 13 at 22:25
• When John's theorem can be applied: yes, the blow up happens in finite time. But if $v$ is unbounded from above (say, has potentially linear growth in time), then in the equations deduced in the answer I wrote above, John's theorem cannot be applied. Apr 14 at 2:43
• @Immanuel John's 1981 paper actually provides also life-span estimates in the small-data limit. If the initial data is size $\epsilon$, Theorem 3 guarantees (in the radial case) solutions exists up to time $T \approx e^{C/\epsilon}$ for some constant $C$. The discussion in section 3 shows that the time of existence cannot exceed $T \approx e^{C'/\epsilon^4}$. The upper bound however depends on the profile of the initial data, and so cannot easily be used to extend Theorem 2. Additionally: the estimates are clearly not sharp (due to the differing powers of $\epsilon$ between the two bounds.) Apr 14 at 13:15


text:
Opened 7 years ago

Closed 3 years ago

# Request ability to set a field as part of workflow

Reported by: Owned by: scottj@… Eli Carter normal AdvancedTicketWorkflowPlugin normal scottj@… 0.11

It would be really nice to be able to set a field as part of my ticket workflow. At the company I work for, we have a workflow that has an analysis stage. During the analysis, the developer assigned reviews the enhancement/defect and determines how much time he/she thinks it will require to implement the changes.

Right now, it's just assumed that the developer will fill in the 'estimatedhours' field themselves, then click the 'analyze' action to transfer it to another state. However, it would be better if the actual action to transfer to another state would read something like, 'analyze at _____ hours", and the developer would type in the number of hours he/she thinks it would require.

So, this request is simply to have some method of embedding a custom field into the action so that the workflow could set a custom ticket field upon making this action. In my example, the code to setup the operation might be:

   analyze = new -> analyzed
analyze.operation = set_field
analyze.field = ticket_custom.estimatedhours
analyze.units = hours

And the resulting actions might be:

• leave as new
• analyze at ______ hours (The estimated hours will be set. Next status will be 'analyzed'.)

+1

### comment:3 Changed 7 years ago by scottj@…

Hi retracile:

Any information on this? Do you know if this could be completed easily?

### comment:4 Changed 6 years ago by Eli Carter

Description: modified (diff)

### comment:5 Changed 3 years ago by Ryan J Ollos

Resolution: → duplicate new → closed

If you want to require a field to be filled in, it is suggested that you use a plugin such as DynamicFieldsPlugin. If you want to set a field to a specific value that is determined from the current state of the workflow, that might eventually be satisfied by a set_field operation suggested in #9502 and trac:#11452. It might be possible to move a field from the ticket properties to the workflow. We certainly don't want it in both places. That will be worked out in the aforementioned tickets.

### Modify Ticket

Change Properties


text:
March 13, 2019

## Plan for Today:

### (2) Correlation

• confounding (again)
• correlation

## Grade Scaling?

### No scaling of grades

• solving "fat tail" would distort upper end of distribution

### Option to Re-Weight

• Commitment NOW to re-weight midterm to final (no "take-backs")
• In principle, open to ANYONE

## Evidence for Causality

#### Internal Validity

A research design has internal validity when the observed relationship between $$X$$ and $$Y$$ it finds is not a biased (systematically incorrect) estimate of the causal effect of $$X$$ on $$Y$$ (does not suffer from confounding).

#### External Validity:

Research design has external validity when the $$X$$ and $$Y$$ we examine match the causal theory and the cases we study match the set of cases/population the causal theory is supposed to describe

• If our study suffers from sampling bias $$\to$$ lack of external validity
• If our independent variable/dependent variables do not match the theory $$\to$$ lack of external validity

## Beyond Experiments:

Recall what hypotheses say: If $$X$$ causes $$Y$$

We should observe that as $$X$$ changes, the potential outcomes of $$Y$$ change.

But FPCI says we can only ever see one potential outcome per case.

We always examine the relationship between the observed $$X$$ and the observed $$Y$$

• true in experiments
• true in any kind of causal investigation

## Beyond Experiments:

To infer a causal relationship between $$X$$ and $$Y$$ based on relationship between the observed $$X$$ and the observed $$Y$$, one of the following must be true

1. potential outcomes of $$Y$$ (the way cases behave when exposed to different levels of $$X$$) are the same for cases with different values of $$X$$
2. (identically) other factors that affect $$Y$$ (call them $$W,Z,etc.$$) are the same across/unrelated to different levels of $$X$$ (this is saying we have an absence of confounding) in the cases we compare.

## Beyond Experiments:

All "solutions" to the FPCI make assumptions about the cases we compare that allow us to accept one of these two points (previous slide), so we can infer causality from how observed values of $$X$$ and $$Y$$ are related.

• Experiments only look at observed values of $$X$$ and $$Y$$, but we can claim causality because we assume random assignment

## Beyond Experiments: Digression

We want to know whether having fewer guns reduces suicide, accidents, and aggressive uses of guns.

Imagine the following experiment:

We collect data on legal gun owners. We randomly assign half of them to receive a letter from the government that offers to pay them money to hand over their guns (a gun buyback). The other half receives no letter. We then compare suicide rates, accident rates, and gun victimization rates among the friends and families of people who handed over their guns against people who did not hand over their guns. We observe that suicide, accident, and gun victimization rates were lower in the "give-up-guns" group. Can we infer that having fewer guns caused a reduction?

## Correlation

#### correlation:

degree of association or relationship between the observed values taken by two variables ($$X$$ and $$Y$$)

• Many different ways of doing this (compare group means, regression) are all fundamentally about correlation.
• correlations have a direction:
• positive: implies that as $$X$$ increases, $$Y$$ increases
• negative: $$X$$ increases, $$Y$$ decreases
• correlations have strength (has nothing to do size of effect):
• strong: $$X$$ and $$Y$$ almost always move together
• weak: $$X$$ and $$Y$$ do not move together very much
• There is also a technical definition of correlation (later)

But with some assumptions: correlation $$\to$$ causation

## Correlation

To understand assumptions, need to know what problems arise

### Two types of problems

• bias (spurious correlation, confounding): $$X$$ and $$Y$$ are correlated but the correlation does not result from causal relationship between those variables

• random association: correlations between $$X$$ and $$Y$$ occur by chance and do not reflect

## Confounding (again)

confounding occurs when some other variable $$W$$ is causally linked to $$X$$ (independent variable) and $$Y$$ (dependent variable).

#### or if we diagram

If we diagram causal links between variables using this notation: $$X \to Y$$ implies $$X$$ causes $$Y$$, then…

confounding occurs when there is a path between $$X$$ and $$Y$$ that is non-causal (goes the "wrong way" on at least one arrow)

## Confounding (again)

Do all third variables create confounding/bias?

### No.

$$W$$ produces no confounding under the following conditions:

1. $$W$$ is unrelated to $$X$$ or $$Y$$
2. $$W$$ is an antecedent variable
3. $$W$$ is an intervening variable

## Confounding (again)

intervening variable: a variable through which $$X$$ causes $$Y$$

• Intervening variables do not produce spurious correlations

$X \rightarrow W \rightarrow Y$

## Antecedent Variable

antecedent variable: a variable that affects $$Y$$ only through $$X$$

• antecedent variables do not produce spurious correlations if they only affect $$Y$$ through $$X$$

$Z \rightarrow X \rightarrow Y$

$$Z$$ is intervening variable


text:
# Restriction of scalars in fibered linear categories

Let $X$ be a fibered linear category over $(Fields/k)^{op}$. This means that for every field extension $K/k$, we have a $K$-linear category $X(K)$ of objects "defined over $K$", whose homomorphisms form finite dimensional vector spaces, and that we may "extend scalars", i.e. for every field extension $L/K$ and every object $M\in X(K)$, we have some $M_L$ satisfying a pullback axiom.

For example, fix a finite dimensional algebra $A$ over $k$, then $X(K)$ might be the category of finite dimensional representations of $A\otimes_kK$ and pullback is given by tensor product.

In all the examples I am interested in, there is this additional property: if an extension $L/K$ is finite and $M\in X(L)$, I can pushforward $M$ to an object of $X(K)$.

Of course this does not follow from the axioms, consider for example the category of representations of modules of some fixed rank.

Is this condition axiomatized in some way? Or are there any other assumptions on the categories $X(K)$ (e.g. abelian?) that might imply it?

• The nice condition is that extension of scalars always has a right adjoint. If the $X(K)$ are sufficiently nice (e.g. presentable) this is true iff extension of scalars is cocontinuous. – Qiaochu Yuan Sep 27 '17 at 22:54


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# Using the average wage of $200 a day to value workers time,  Question banks Downloads My Bookmarks Reviews Important topics Author Message TAGS: Manager Joined: 09 Feb 2013 Posts: 120 Followers: 1 Kudos [?]: 549 [0], given: 17 Using the average wage of$200 a day to value workers time, [#permalink]  08 Jun 2013, 00:29
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Using the average wage of $200 a day to value workers’ time, and assuming that people who could ask more questions online would, the analyst estimated the search engine’s worth might be approximately$300 annually to the average worker.
A)Using the average wage of $200 a day to value workers’ time, and assuming that people who could ask more questions online would, the analyst estimated the search engine’s worth might be$300 annually to the average worker.
B)Using the average wage of $200 a day as a value of workers’ time and the assumption that people who could ask more questions online would, the analyst estimated the search engine’s worth might be$300 annually to the average worker.
C)Assuming the average wage of $200 a day as a value of workers’ time and that people who could ask more questions online would, the analyst estimated that the search engine might be worth$300 annually to the average worker.
D)Assuming the average wage of $200 a day as a value of workers’ time and the fact that people who could ask more questions online would, the analyst estimated that the search engine might be worth$300 annually to the average worker.
E)Using the average wage of $200 a day as a value of workers’ time, and assuming that people who could ask more questions online would, the analyst estimated that the search engine might be worth$300 annually to the average worker.
[Reveal] Spoiler: OA

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Re: Using the average wage of $200 a day to value workers’ tim [#permalink] 08 Jun 2013, 10:57 B,C,D are out because of //ism B:- using(simple gerund)not // to assumption(noun) C:-that clause //ism issue D:-that clause //ism issue but dont know how to eliminate A...is it because of poosesive case(search engine’s ).. pls help.. Manager Status: Working, Small Business Owner & 1st GMAT 750 Joined: 01 Dec 2012 Posts: 89 Location: United States (OH) Concentration: General Management, Social Entrepreneurship GMAT 1: 750 Q50 V42 GPA: 2.9 WE: Analyst (Consulting) Followers: 6 Kudos [?]: 65 [0], given: 63 Re: Using the average wage of$200 a day to value workers’ tim [#permalink]  08 Jun 2013, 11:08
I chose option E.

We could eliminate C & D because of its use of "Assuming the ____ as __ " which is incorrect.

Between A, B & E - Only E correctly uses the form "the analyst estimated that the" instead of "the analyst estimated ...."

Does this help explain it?
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Article --> http://www.topmba.com/blog/could-mba-de ... ay-forward

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Re: Using the average wage of $200 a day to value workers’ tim [#permalink] 08 Jun 2013, 22:21 1 This post received KUDOS vishnuns39620 wrote: I chose option E. We could eliminate C & D because of its use of "Assuming the ____ as __ " which is incorrect. Between A, B & E - Only E correctly uses the form "the analyst estimated that the" instead of "the analyst estimated ...." Does this help explain it? Hi vishnuns, can you please elaborate more why you think that "Assuming the ____ as __ " and "the analyst estimated ...." are wrong? About the second construct in particular, I think that is perfectly legit: the analyst estimated the search engine’s worth Why do you think that this kind of strucuture is wrong? Whenever you see verbs like: "say/tell/suggest/announce/estimate/..." most of the times "that" is required, but do not take this as a Rule of Thumb. Example: The computer company has announced that it will purchase (...). In this sentence that is required and refers to the entire clause. But when you have other types of counstruct as "analysts estimated the worth", "that" is not required. "The worth" is the actual thing that he estimated, so the phrase without "that" is correct. I hope that what I mean is clear _________________ It is beyond a doubt that all our knowledge that begins with experience. Kant , Critique of Pure Reason Tips and tricks: Inequalities , Mixture | Review: MGMAT workshop Strategy: SmartGMAT v1.0 | Questions: Verbal challenge SC I-II- CR New SC set out !! , My Quant Rules for Posting in the Verbal Forum - Rules for Posting in the Quant Forum[/size][/color][/b] Current Student Joined: 06 Sep 2013 Posts: 2036 Concentration: Finance GMAT 1: 770 Q0 V Followers: 39 Kudos [?]: 432 [1] , given: 355 Re: Using the average wage of$200 a day to value workers time, [#permalink]  07 Oct 2013, 08:44
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emmak wrote:
Using the average wage of $200 a day to value workers’ time, and assuming that people who could ask more questions online would, the analyst estimated the search engine’s worth might be approximately$300 annually to the average worker.
A)Using the average wage of $200 a day to value workers’ time, and assuming that people who could ask more questions online would, the analyst estimated the search engine’s worth might be$300 annually to the average worker.
B)Using the average wage of $200 a day as a value of workers’ time and the assumption that people who could ask more questions online would, the analyst estimated the search engine’s worth might be$300 annually to the average worker.
C)Assuming the average wage of $200 a day as a value of workers’ time and that people who could ask more questions online would, the analyst estimated that the search engine might be worth$300 annually to the average worker.
D)Assuming the average wage of $200 a day as a value of workers’ time and the fact that people who could ask more questions online would, the analyst estimated that the search engine might be worth$300 annually to the average worker.
E)Using the average wage of $200 a day as a value of workers’ time, and assuming that people who could ask more questions online would, the analyst estimated that the search engine might be worth$300 annually to the average worker.

Wow! This one was tough took me almost 2 min! Chose (E). In future posts it would definetely be advisable to leave some space between answer choices, otherwise I get the impression it is harder than it really is.

Now back to the question, parallelism in E is correct - Using // assuming. Looking good.
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Re: Using the average wage of $200 a day to value workers time, [#permalink] 07 Oct 2013, 11:27 1 This post received KUDOS I selected E over A because "might be approximately" is redundant. Intern Joined: 04 Dec 2011 Posts: 29 Followers: 0 Kudos [?]: 6 [0], given: 6 Re: Using the average wage of$200 a day to value workers time, [#permalink]  16 Oct 2013, 17:27
emmak wrote:
Using the average wage of $200 a day to value workers’ time, and assuming that people who could ask more questions online would, the analyst estimated the search engine’s worth might be approximately$300 annually to the average worker.
A)Using the average wage of $200 a day to value workers’ time, and assuming that people who could ask more questions online would, the analyst estimated the search engine’s worth might be$300 annually to the average worker. <<Estimated That ---That missing >>
B)Using the average wage of $200 a day as a value of workers’ time and the assumption that people who could ask more questions online would, the analyst estimated the search engine’s worth might be$300 annually to the average worker.<<Estimated That ---That missing >>
C)Assuming the average wage of $200 a day as a value of workers’ time and that people who could ask more questions online would, the analyst estimated that the search engine might be worth$300 annually to the average worker. <<Meaning change.It should be using the average wage>>
D)Assuming the average wage of $200 a day as a value of workers’ time and the fact that people who could ask more questions online would, the analyst estimated that the search engine might be worth$300 annually to the average worker. <<Meaning change.It should be using the average wage and the usage of fact.which is redundant >>
E)Using the average wage of $200 a day as a value of workers’ time, and assuming that people who could ask more questions online would, the analyst estimated that the search engine might be worth$300 annually to the average worker.
<<Correct..Correct usage of assuming that >>
Please correct me If I am wrong.The usage of fact is not correct?Can any one tell me is it always incorrect to use fact as a key word in SC.
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Re: Using the average wage of $200 a day to value workers time, [#permalink] 07 Jan 2014, 05:22 1 This post received KUDOS Hi, You estimate THAT ==> A and B are out! C and D are wordy ==> left with E! Hope it helps _________________ Think outside the box Manager Status: Work hard in silence, let success make the noise Joined: 25 Nov 2013 Posts: 161 Location: India Concentration: Finance, General Management GMAT 1: 540 Q50 V15 GMAT 2: 640 Q50 V27 GPA: 3.11 WE: Consulting (Computer Software) Followers: 6 Kudos [?]: 52 [0], given: 84 Re: Using the average wage of$200 a day to value workers time, [#permalink]  07 Jan 2014, 23:56
B,C and D are incorrect because they lack parallelism.
Out of A and E, E uses "estimated that", whereas A doesn't.

So, the correct answer is E.
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Re: Using the average wage of $200 a day to value workers time, [#permalink] 08 Jan 2014, 00:45 emmak wrote: Using the average wage of$200 a day to value workers’ time, and assuming that people who could ask more questions online would, the analyst estimated the search engine’s worth might be approximately $300 annually to the average worker. A)Using the average wage of$200 a day to value workers’ time, and assuming that people who could ask more questions online would, the analyst estimated the search engine’s worth might be $300 annually to the average worker. B)Using the average wage of$200 a day as a value of workers’ time and the assumption that people who could ask more questions online would, the analyst estimated the search engine’s worth might be $300 annually to the average worker. C)Assuming the average wage of$200 a day as a value of workers’ time and that people who could ask more questions online would, the analyst estimated that the search engine might be worth $300 annually to the average worker. D)Assuming the average wage of$200 a day as a value of workers’ time and the fact that people who could ask more questions online would, the analyst estimated that the search engine might be worth $300 annually to the average worker. E)Using the average wage of$200 a day as a value of workers’ time, and assuming that people who could ask more questions online would, the analyst estimated that the search engine might be worth $300 annually to the average worker. I think A would have been correct, had it been the analyst estimated that the. So, E is correct in this case. What do you think, fellas? Intern Joined: 20 May 2013 Posts: 10 Location: India GMAT 1: 550 Q47 V20 GPA: 3.7 WE: Other (Computer Software) Followers: 1 Kudos [?]: 5 [0], given: 42 Re: Using the average wage of$200 a day to value workers’ tim [#permalink]  14 Jan 2014, 06:08
Zarrolou wrote:
vishnuns39620 wrote:
I chose option E.

We could eliminate C & D because of its use of "Assuming the ____ as __ " which is incorrect.

Between A, B & E - Only E correctly uses the form "the analyst estimated that the" instead of "the analyst estimated ...."

Does this help explain it?

Hi vishnuns,

can you please elaborate more why you think that "Assuming the ____ as __ " and "the analyst estimated ...." are wrong?

About the second construct in particular, I think that is perfectly legit:
the analyst estimated the search engine’s worth
Why do you think that this kind of strucuture is wrong?

Whenever you see verbs like: "say/tell/suggest/announce/estimate/..." most of the times "that" is required, but do not take this as a Rule of Thumb.
Example:
The computer company has announced that it will purchase (...). In this sentence that is required and refers to the entire clause.

But when you have other types of counstruct as "analysts estimated the worth", "that" is not required.
"The worth" is the actual thing that he estimated, so the phrase without "that" is correct.

I hope that what I mean is clear

Hi Zarrolou,

If option A is not wrong for this reason, then what's wrong with this option? I think the difference between A and E has more to do with this sentence fragment: "$200 as a value of worker's time". Please help. VP Joined: 08 Jun 2010 Posts: 1057 Followers: 1 Kudos [?]: 72 [0], given: 328 Re: Using the average wage of$200 a day to value workers time, [#permalink]  03 Sep 2015, 17:33
if the second part of ellipsis contains helping verb, the first part must contain the same helping verb, possibly in different tense. if the case is not like this, we write down full verb phrase in the second part.

I think in E , "would " should be " would ask"


text:
H_2(g) + 2NO(g) -> H_20(g) + N_2O(g). At high temperatures, doubling the concentration of H_2 doubles the rate of reaction, while doubling the concentration of NO increases the rate fourfold. How do you write a rate law for this reaction?

Aug 14, 2017

We gots ${H}_{2} \left(g\right) + 2 N O \left(g\right) \rightarrow {H}_{2} O \left(g\right) + {N}_{2} O \left(g\right)$

Explanation:

Now $\text{rate}$ $= k {\left[{H}_{2}\right]}^{x} {\left[N O\right]}^{y}$, where $x$ and $y$ are exponents, whose magnitude we have to assess by experiment.

So ${\text{rate}}_{1} = k {\left[{H}_{2}\right]}^{x} {\left[N O\right]}^{y}$

So ${\text{rate"_2=k[2H_2]^x[NO]^y=2xx"rate}}_{1}$, clearly $x = 1$

But ${\text{rate"_3=k[H_2][2NO]^y=4xx"rate}}_{1}$, clearly $y = 2$.

And we write the overall rate law.....$\text{rate} = k \left[{H}_{2}\right] {\left[N {O}_{2}\right]}^{2}$

Note that this conveniently follows the stoichiometric equation given; it did not NECESSARILY have to follow this rate law.


text:
## Calculus: Early Transcendentals 8th Edition

a) $\frac{dP}{dt}$ represents the percentage rate of change per year. b) $\frac{dP}{dt}|_{t=2}$ represents the percentage rate of change in the city's electrical power after 2 years.
a) $\frac{dP}{dt}$ represents the percentage rate of change in the city's electricity power that is produced by solar panels since 2000. b) $\frac{dP}{dt}|_{t=2} = 3.5$ represents the percentage rate of change of the city's electrical power produced by solar panels in the year 2002.


text:
# An indefinite Integral Problem with algebric numerator and trigonometric denominator

$$\int \frac{x^2+(n(n-1))}{(x\sin x +n\cos x)^2 } dx$$ I know this is an homework problem, but I really couldn't think of any way to solve it. Like DI Method (No go) , What kind of substitution as denominator is trigonometric whereas Numerator is algebric. Thought of n(n-1) can come by double differentiating but.. like how would we have it here ... etc confusing and weird thoughts. Please help me out

• It is $$\frac{n \sin (x)-x \cos (x)}{n \cos (x)+x \sin (x)}+C$$ – Dr. Sonnhard Graubner May 19 at 6:10
• @Dr.SonnhardGraubner How ?? Please show /tell the working and thought process – user232243 May 19 at 6:16
• Integration by parts might help, after some manipulation. – arya_stark Jun 19 at 8:28

$$I=\int\frac{x^2+n(n-1)}{(x\sin x+n\cos x)^2}dx$$

Put $$x=n\tan \theta\;\;dx=n\sec^2\theta d\theta$$

$$I=\int\frac{n^2\tan^2\theta+n^2-n}{(n\tan\theta\sin(n\tan \theta)+n\cos(n\tan \theta))^2}\cdot n\sec^2\theta d\theta$$

$$I=\int\frac{n\sec^2(\theta)-1}{\cos^2(n\tan\theta-\theta)}d\theta.$$

Put $$n\tan \theta-\theta=u$$ and $$(n\sec^2\theta-1)d\theta=du$$

$$I=\int\frac{1}{\cos^2u}du=\int\sec^2(u)du=\tan u+C$$

$$I=\tan(n\tan\theta-\theta)+C=\tan\bigg(x-\tan^{-1}\frac{x}{n}\bigg)+C$$

$$I=\frac{n\sin x-x\cos x}{x\sin x+n\cos x}+C$$

• Thank You so much. If you don't mind me asking , what was your thought process , behind it that you could think of substitution of $x=n*tan(\theta)$ – user232243 May 20 at 3:44
• From Dr. Answer...... – jacky May 20 at 8:29

$$I = \int\frac{x^2 + n(n-1)}{(x\sin x + n\cos x )^2}dx$$

Now we'll try to convert it into the form of $$\frac{a}{y} + \frac{b}{y^2}$$, where $$a,b$$ are functions of $$x$$ and $$y$$ is the denominator.

$$x^2+n(n-1) =(x\sin x + (n-1)\cos x)(x\sin x+n\cos x)-((1-n)\sin x + x\cos x)(n\sin x - x\cos x)$$

$$I = \int \big[\frac{(x\sin x + (n-1)\cos x)(x\sin x+n\cos x)}{(x\sin x + n\cos x )^2} - \frac{((1-n)\sin x + x\cos x)(n\sin x - x\cos x)}{(x\sin x + n\cos x )^2}\big]dx$$

Now, $$I = \int\big[\frac{(x\sin x + (n-1)\cos x)}{(x\sin x + n\cos x )} - \frac{((1-n)\sin x + x\cos x)(n\sin x - x\cos x)}{(x\sin x + n\cos x )^2}\big]dx$$

Let $$I_1 = \int\frac{(x\sin x + (n-1)\cos x)}{(x\sin x + n\cos x )}dx$$ , $$I_2 = \frac{((1-n)\sin x + x\cos x)(n\sin x - x\cos x)}{(x\sin x + n\cos x )^2}dx$$

In $$I_2$$,

let $$u = n\sin x - x\cos x$$, $$dv = \frac{(1-n)\sin x + x\cos x}{(x\sin x + n\cos x )^2}dx$$

$$du = (n\cos x - \cos x + x\sin x)dx$$ ,

[In $$v$$ , $$t = x\sin x + n\cos x$$, $$dt = (x\cos x + \sin x - n\sin x )dx = (x\cos x + (1-n)\sin x)dx$$]

$$v = \int\frac{(1-n)\sin x + x\cos x}{(x\sin x + n\cos x )^2}dx = \int\frac{dt}{t^2} = -\frac{1}{t} = - \frac{1}{x\sin x + n\cos x}$$

So,

$$I_2 = uv - \int vdu = -(n\sin x - x\cos x)\frac{1}{x\sin x + n\cos x} + \int (n\cos x - \cos x + x\sin x).\frac{1}{x\sin x + n\cos x}dx + c$$

$$I_2 = - \frac{n\sin x - x\cos x}{x\sin x + n\cos x} + \int \frac{x\sin x + (n-1)\cos x}{x\sin x + n\cos x}dx +c$$

$$I_2 = - \frac{n\sin x - x\cos x}{x\sin x + n\cos x} + I_1 +c$$

$$I = I_1 - I_2 = \frac{n\sin x - x\cos x}{x\sin x + n\cos x} + k$$

($$k = -c$$)


text:
# Spectral radius sum of two matrices

Let $$W = S + cT$$, where $$|c| \le 1$$ is a real constant and where $$S$$ and $$T$$ are square matrices containing real numbers from the interval $$[0,1]$$.

Assume moreover that the all eigenvalues $$\lambda$$ of $$S$$ and $$T$$ satisfy $$|\lambda| \le 1$$. (Both matrices are right stochastic in that each row sums to one.)

Is it possible to derive an informative bound for the spectral radius of $$W$$? Besides being right stochastic, the matrices $$S$$ and $$T$$ do not have a special structure (e.g. Hermitian, diagonalizable, ...).

• Is en.wikipedia.org/wiki/Weyl%27s_inequality sufficient for you? – Sandeep Silwal Nov 8 at 18:51
• @SandeepSilwal: According to the OP, the matrices $S$ and $T$ are not assumed to be Hermitian. – Jochen Glueck Nov 8 at 18:58
• Welcome to MathOverflow! I edited the wording of the question since the expression "weakly smaller than" for a non-strict inequality seems to be very uncommon. – Jochen Glueck Nov 8 at 19:04
• Concerning the content of the question, do you assume that $c \ge 0$? – Jochen Glueck Nov 8 at 19:05
• @JochenGlueck, thank you for your comment. I have eddited my orginal question, I hope it is now more clear. – Seb Nov 10 at 14:07


text:
Quantum computations, irreversibility, unitarity

1. Mar 3, 2008

kdv

A stupid couple of questions...

In quantum computations, one typically starts with some initial quantum state on which an operator is applied.

This operator must be unitary , right? (I guess that otherwise, it would not corerspond to an actual physical quantum setup). And this implies that the operator must be reversible, correct?
So in quantum computing, we consider only irreversible processes, correct?

Now consider Deutsch's algorithm. It deals with a function acting on a single qubit which may be either "balanced" or "constant".

It is balanced if it may give both 0 and 1 a sresult. One possibility is f(0)=0, f(1)=1 and the other balanced possibility is f(0)=1, f(1)=0.

It is constant if it gives the samr answer no matter what the input is. So the first possibility is f(0) =f(1) =0 and the second possibility is f(0)=f(1)=1.

Now, the whole idea of Deutsch's algorithm is to able to distinguish whether f is balanced or constant with a single application of a certain operation which must be chosen judiciously. (and the input state must be chosen judiciously as well).

Now, usually people say: classically, one cannot determine if the function is constant or balanced without two runs fo teh function. (for example, observing that f(0) gives zero does not tell us if it's balanced or not). Then people say that quantum mechanically, one can tell if f is balanced or not with a single run of an operator applied to a suitable combination of two qubits.

Fine. But my question is this: if I understand correctly, the function f(x) itself cannot be represented by any quantum operator , correct? Because in the constant case it is not reversible and therefore could not be associated to a unitary operator.

This kind of surprised me when I thought about this today...It's weird in some sense because the starting point is the operator "f" and yet I have never seen anyone mentioning that f itself cannot be implemented as a quantum operation. Maybe I am missing something?

2. Mar 3, 2008

Hurkyl

Staff Emeritus
He probably means something like the following operation on two qubits:

|x>|y> ---> |x>|y + f(x)>

3. Mar 3, 2008

kdv

Yes, I know this is the operation used in Deutsch's algorithm. I was wondering if I was correct in saying that the operation on a single qubit |x> ---> |f(x)> would be impossible to implement.


text:
# 4.5.3 Viskovatov's Algorithm for the General Case

In order to generalize Viskovatov's algorithm to degenerate cases, one must make an inspired choice for the representation of the particular \L/M] Pade aDDroximant as a continued fraction. We consider

which should be compared with the general C-fraction (5.12) and Magnus P-fraction (5.16). In (5.17), each p,(z) is a polynomial with ^(0)^=0 and each a; is a positive integer: each p^z) and a, are fully defined in (5.21) and (5.22). As usual, we suppose that we are given the series expansion

and that we are requested to find the [L/M] Pade approximant of (5.18) with L>M. If L<M, we should remove the factor zk from (5.18), where k is the highest power such that zk factors (5.18), and start with its reciprocal series. We choose to consider an infinite series in (5.18) for convenience of representation; in fact only L + M+ 1 terms of (5.18) are required, and questions of convergence do not arise. Following Werner [1979, 1980], we define the generalized Viskovatov reduction:

We may then define the polynomial Pj(z) of degree strictly less than a by

Initialization. Let

and

Iteration. For j=0,1,2,..., proceed recursively. Define a] to be the least integer for which

Our aim is to find the [/-/w -] Pade approximant of the left-hand side of (5.22). If at >lj + m , then let n=j. In this case, (5.17) is the Pade approximant required, as is proved in Theorem 4.5.1. If

the [L/M] Pade approximant to the given series (5.18) does not exist, as is proved in Theorem 4.5.1.

Next, we consider the reciprocal series defined by

so that, formally at least,

In order to find the [lj/m ] Pade approximant of the left-hand side of (5.25), we seek the [my//y—ay] Pade approximant of the left-hand side of (5.24V Hence we define

and

The requirement that

is guaranteed by (5.21), and also

in view of (5.23V Hence we need to find the

of the series

defined by (5.24) and (5.26), and the process

becomes recursive.

Having defined the iterative process for the construction of (5.17), we must verify that it fulfills its specification. From (5.26), we note that

and therefore

where deg{/?(z)} is the degree of p(z). We see that the fraction (5.17) has at most L + M divisors, which occurs if a,— 1 for / = 0,1,2,..., n, and hence the process terminates.

Theorem 4.5.1. The generalized Viskovatov reduction defined in (5.19)— (5.26) is reliable: if the [L/M] Pade approximant exists, the process finds if, if not, (5.23) is satisfied and indicates this fact.

Proof Suppose that the [L/M] approximant to (5.20) exists, yet the algorithm fails. Let the approximant have the Maclaurin expansion

for

and

derived by the reduction is identical when applied to

itself. Initially, with

we find

so that

Since the reduction process (5.25) for calculating the [L/M] approximant is based on the series-expansion coefficients only, the representation (5.17)

This division of

yields 0n(z) as the quotient anc

as the remainder. Unless on(z) itself is the [L/ Wm] Padé approximant of

/(z), (5.21) ensures that

and so we may use the Maclaurin

expansion

We sunnose that

in (5.31) is reduced sequentially by the

process indicated by (5.19)-(5.26) as far as possible. Let

fory=0,1k. Hence we reduce the Padé approximant sequentially by

where we define

and

Suppose that the process fails at stage k, namely because (5.23) is satisfied with j=k. From (5.33),

where lk >ak >lk +mk. Because the left-hand side of (5.37) has degree at most lk, then either ak *Zlk or else T(k\z) = 0. This is incompatible with a failure at stage k. Consequently the reduction process cannot fail if the [L/M] Padé approximant exists.

The proof of the converse result, that R(0)(z) constructed by (5.17)-(5.26 > is in fact the [L/M] Padé approximant, uses the definitions of these equations. From (5.22) and (5.28), we see that

where

we define

and for

We define

We see directly from (5.40) that

Let us suppose that

It then follows from (5.25) and (5.41) that (5.42) is true with k^k~\. Using (5.39), we have proved that

by induction. This completes the proof of the theorem.

variable

is also a factor ol

but it is obvious that each common factor of

Hence we see

that we may view the reduction process and Theorem 4.5.1 as a proof, using

In the course of the proof we can see the connection between (5.31). (5.331 and the Euclidean algorithm. The connection is direct using the

the Euclidean algorithm, that P(0\z) and (?<0|(z ) have no common factors. Of course, the proof in Theorem 1.4.3 is more direct.

There is also a connection between (5.33) and the recurrence relations (2.14). From (5.33), we find that

for

We may suppose that the relative normalization

is fixed by defining

From (5.35), (5.36), and (5.44), it follows that

and

which are remarkably similar to (5.2.14) when viewed as a recurrence for j=n—\,n — 2,...,l. In view of (5.34), (5.41), and (5.45), the recurrence defines the numerator and denominator of

However, the recurrence

(5.48)

initialized by (5.45), is probably the more convenient

form.

It is now clear how to define a modified Viskovatov algorithm, based on (5.17)-(5.26) and incorporating a generalization of (3.1) for efficiency. In fact, we may as well consider the problem of finding an [L/M] Padé approximant, with L>M, for

In (5.49), the coefficients gj°\ dj0) for / = 0.1,2           L + M are given, and

^ 7^0. This problem is solved using bigradients in Section 1.6. We outline a generalized Viskovatov algorithm, emphasizing only the alterations to (5.17)—(5.26) [Bultheel, 1980b],

Initialization. Define l0—L, m0=M.

Recurrence. The process is iterative for7 = 0,1,2,.... We construct the polynomial pj{z) of degree at most lJ — mJ such that

(5.50)

with a ■ > lj —mr In fact, we need to construct the coefficients of

(5.511

coefficients of the series

(5.521

The construction of the coefficients of (5.51) and (5.52) is performed iteratively, according to the algebra implied by the following analysis. Let

For k = 0,1,...,/—m define

and

HencePj(z) is specified by (5.51), and for the coefficients of (5.52) we have

(5.53)

We define a; as the least integer such that (5.53) holds with h\tn 7^0, and hence the coefficients of (5.52) are fully specified.

If

Otherwise, let

or if (5.53) vanishes exactly, the iteration is terminated

the Padé approximant required is nonexistent.

and

for

In this case,

the /th stage of the reduction

process (5.54) may be iterated. The algorithm is now fully specified.

The generalized Viskovatov algorithm in (5.49)-(5.54) is designed to yield an [L/M] Padé approximant reliably. It follows a path on the Padé table which avoids nonexistent entries in blocks in the table. We conclude that the path that the convergents of the fraction (5.17) follows in the Padé table depends on the values of the coefficients in (5.49) as well as the values of L and M. As a computational algorithm, it can be recommended wherever exact arithmetic is available and the values of a0, a,, ...,«„_, are unambiguously defined. The techniques of the generalized Viskovatov algorithm can be adapted to rational interpolation using a generalized Thiele fraction (see Part II, Section 1.1, and Werner [ 1979a, b]) except that the algorithm must incorporate a facility for reordering the interpolation points if necessary [Thacher and Tukey, 1960; Graves-Morris and Hopkins, 1978; Graves- Morris, 1980a], Similar techniques based on the Euclidean algorithm allow Kronecker's algorithm to be generalized to avoid inessential degeneracies (see Section 2.4; Graves-Morris [1980a]).


text:
Question: Consider the following sequence of 5 numbers: 0.15, 0.94, 0.05, 0.51, 0.29.
0

Subject: Computer Simulation and Modeling

Topic: Random number generation

Difficulty: Low

### Consider the following sequence of 5 numbers: 0.15, 0.94, 0.05, 0.51, 0.29. Use the Kolmogorov-Smirnov test with α = 0.05 to determine if the hypothesis that the numbers are uniformly distributed on the interval [0, 1] can be rejected. Use D0.05, 5 = 0.565.

csm(28) • 170 views


text:
Approximation numbers of certain Volterra integral operators.(English)Zbl 0658.47049

This paper studies a Volterra integral operator of the form $Tf(x)=v(x)\int^{x}_{0}u(t)f(t)dt\quad (x\in {\mathbb{R}}^+=[0,\infty)),$ whose kernel satisfies only local integrability conditions.
The authors give necesary and sufficient conditions for the boundedness and compactness of T.
An objective of the paper is to obtain upper and lower bounds for the limit of the approximation numbers of T, which maps $$L^ p({\mathbb{R}}^+)$$ to $$L^ q({\mathbb{R}}^+).$$
By way of illustration it is shown that, when $$u(x)=e^{Ax}$$, $$v(x)=e^{-Bx}$$ with $$B>A>0$$ and $$p=q\in (1,\infty)$$ then the mth approximation number of T is bounded above and below by multiples of $$m^{-1}$$.
Reviewer: L.Hącia

MSC:

 47Gxx Integral, integro-differential, and pseudodifferential operators 47B06 Riesz operators; eigenvalue distributions; approximation numbers, $$s$$-numbers, Kolmogorov numbers, entropy numbers, etc. of operators 45P05 Integral operators
Full Text:


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## Proof: (related to "Uniqueness of the Empty Set")

• For any two given non-empty sets $X,Y$ assume, their elements $x\in X,y\in Y$ are sets themselves.
• By the extensionality principle they are equal $x=y$ if they have the same elements.
• This argument can be repeated for the elements of these elements (if any). We can repeat this argument for all elements of elements of elements, etc.
• At some point, we might arrive at elements $x\in X$ and $y\in Y$ which do not contain any elements. Note that the existence of such empty elements $x=\emptyset$ and $y=\emptyset$ is ensured by the axiom of empty set.
• Now, the extensionality principle says again, that $x=y$ are equal, even though $x=\emptyset$ and $y=\emptyset$.
• In other words, all empty sets are not distinguishable. Thus, $\emptyset$ is unique.
q.e.d

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# Needs explanation for the solution of question relating to the area of ellipse

• MHB
• WMDhamnekar
WMDhamnekar
MHB
Question:- Find the area of the figure given by the cartesian equation below:

$$\frac{(x+y)^2}{16}+\frac{(x-y)^2}{9}=1$$

Solution given:-

Let $x+y= 4\cos{\alpha},x-y=3\sin{\alpha}$ Then $x=\frac{4\cos{\alpha}+3\sin{\alpha}}{2}$ $\Rightarrow dx=\frac{3\cos{\alpha}-4\sin{\alpha}}{2}d\alpha$

$y=\frac{4\cos{\alpha}-3\sin{\alpha}}{2}$. So, $ydx=\left(3-\frac{25}{8}*\sin{2\alpha}\right)*d\alpha$ What is this ydx?

Hence the required area is

$$\displaystyle\int_0^{2\pi}\left(3-\frac{25}{8}*\sin{2\alpha}\right)*d\alpha$$ Why does the author select ydx as integrand for computation of area of ellipse? What is the logic behind that?

$=6\pi=18.849$

If any member of Math help Board knows the explanations for my queries, may reply to this question.

## Answers and Replies

Homework Helper
MHB
$y$ is the y-coordinate of a point on the edge of the figure. $dx$ is an infinitely small increment of the x-coordinate.
The product $y\,dx$ represents a rectangle of height $y$ and width $dx$. Its area is $y\,dx$.
If we add all such rectangles together, we get the area of the figure.

It's actually more or less the definition of an integral.
See here how that interpretation works.


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# Formula for sound dissipation in air?

1. Feb 28, 2012

### Ralphonsicus

Is there an equation that deals with the dissipation of sound waves in air? As in, how much they die out with distance, because we know they must, otherwise you could hear me talking now!

2. Feb 28, 2012

### Hobold

The intensity is inversely proportional to the radius squared.

That is, if a certain source has a power P and it emits pressure waves spherically, the intensity I you'll hear is

$$I = \frac{P}{4\pi r^2}$$

3. Feb 29, 2012

### sophiecentaur

There is also some absorption, which depends on frequency (proportional to frequency squared) and humidity. This produces an attenuation which is proportional to distance (as opposed to inverse square law). Both effects occur at the same time.


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Question

# A sphere of diameter 18 cm is dropped into a cylindrical vessel of diameter 36cm, partly filled with water. If the sphere is completely submerged, then the water level rises (in cm) by

A
3
B
4
C
5
D
6

Solution

## The correct option is A $$3$$Let $$r$$ & $$R$$ be the radius of the sphere and cylindrical vessel respectively.Now,  $$2r=18cm\Rightarrow r=9cm$$         $$2R=36cm\Rightarrow R=18cm$$Let the rise in water level in the cylindrical vessel be $$'h'$$ cm.Volume of sphere $$=\dfrac { 4 }{ 3 } \pi { r }^{ 3 }$$Volume of liquid displaced in the cylindrical vessel $$=\pi { R }^{ 2 }h$$If the sphere is completely submerged in the vessel, then volume of liquid displace in the cylindrical vessel $$=$$ Volume of the sphere$$\therefore \quad \pi { R }^{ 2 }h=\dfrac { 4 }{ 3 } \pi { r }^{ 3 }\Rightarrow { \left( 18 \right) }^{ 2 }\times h=\dfrac { 4 }{ 3 } \times { \left( 9 \right) }^{ 3 }$$$$\Rightarrow \quad h=\dfrac { 4\times { \left( 9 \right) }^{ 3 } }{ 3\times { \left( 18 \right) }^{ 2 } } =3cm$$Thus, the rise in water level in the cylindrical vessel is $$3cm.$$Mathematics

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BUCKET-SORT(A)
1      let B[0...n-1] be a new array
2      n = A.length
3      for i – 0 to n – 1
4      make B[i] an empty list
5      for i = 1 to n
6      insert A[i] into list B[nA[i]]
7      for i = 0 to n – 1
8      sort list B[i] with insertion sort
9      concatenate the lists B[0] , B[1] , ….,B[n-1] together in order

illustrate the operation of BUCKET-SORT on the array $A=\langle .79,.13,.16,.64,.39,.20,.89,.53,.71,.42\rangle$

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